Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3

Percentage Accurate: 68.8% → 87.4%

Time: 12.3s

Alternatives: 17

Speedup: 0.8×

Specification

Math

\[\begin{array}{l} \\ x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y x) (- z t)) (- a t))))

C

double code(double x, double y, double z, double t, double a) {
	return x + (((y - x) * (z - t)) / (a - t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (((y - x) * (z - t)) / (a - t))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + (((y - x) * (z - t)) / (a - t));
}

Python

def code(x, y, z, t, a):
	return x + (((y - x) * (z - t)) / (a - t))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + (((y - x) * (z - t)) / (a - t));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y x) (- z t)) (- a t))))

C

double code(double x, double y, double z, double t, double a) {
	return x + (((y - x) * (z - t)) / (a - t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (((y - x) * (z - t)) / (a - t))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + (((y - x) * (z - t)) / (a - t));
}

Python

def code(x, y, z, t, a):
	return x + (((y - x) * (z - t)) / (a - t))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + (((y - x) * (z - t)) / (a - t));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 87.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y - \frac{y - x}{t} \cdot \left(z - a\right)\\ t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-291}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;y - \left(-x\right) \cdot \frac{z - a}{t}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+298}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- y (* (/ (- y x) t) (- z a))))
        (t_2 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -1e-291)
       t_2
       (if (<= t_2 0.0)
         (- y (* (- x) (/ (- z a) t)))
         (if (<= t_2 2e+298) t_2 t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y - (((y - x) / t) * (z - a));
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -1e-291) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = y - (-x * ((z - a) / t));
	} else if (t_2 <= 2e+298) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y - (((y - x) / t) * (z - a));
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= -1e-291) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = y - (-x * ((z - a) / t));
	} else if (t_2 <= 2e+298) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y - (((y - x) / t) * (z - a))
	t_2 = x + (((y - x) * (z - t)) / (a - t))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= -1e-291:
		tmp = t_2
	elif t_2 <= 0.0:
		tmp = y - (-x * ((z - a) / t))
	elif t_2 <= 2e+298:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y - Float64(Float64(Float64(y - x) / t) * Float64(z - a)))
	t_2 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -1e-291)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(y - Float64(Float64(-x) * Float64(Float64(z - a) / t)));
	elseif (t_2 <= 2e+298)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y - (((y - x) / t) * (z - a));
	t_2 = x + (((y - x) * (z - t)) / (a - t));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= -1e-291)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = y - (-x * ((z - a) / t));
	elseif (t_2 <= 2e+298)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y - N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -1e-291], t$95$2, If[LessEqual[t$95$2, 0.0], N[(y - N[((-x) * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+298], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -inf.0 or 1.9999999999999999e298 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 38.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 57.0

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 57.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z - t}{a}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z - t}{a}}, x\right) \]

      6. lower--.f64 52.0

         \[\leadsto \mathsf{fma}\left(y - x, \frac{\color{blue}{z - t}}{a}, x\right) \]

   7. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

   8. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   9. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(y - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{z \cdot \left(y - x\right)}{t}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      3. metadata-eval N/A

         \[\leadsto \left(y - \color{blue}{1} \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      4. *-lft-identity N/A

         \[\leadsto \left(y - \color{blue}{\frac{z \cdot \left(y - x\right)}{t}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      5. metadata-eval N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      6. *-lft-identity N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{\frac{a \cdot \left(y - x\right)}{t}} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{y - \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      8. div-sub N/A

         \[\leadsto y - \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      9. lower--.f64 N/A

         \[\leadsto \color{blue}{y - \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      10. div-sub N/A

          \[\leadsto y - \color{blue}{\left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      11. associate-/l* N/A

          \[\leadsto y - \left(\color{blue}{z \cdot \frac{y - x}{t}} - \frac{a \cdot \left(y - x\right)}{t}\right) \]

      12. associate-/l* N/A

          \[\leadsto y - \left(z \cdot \frac{y - x}{t} - \color{blue}{a \cdot \frac{y - x}{t}}\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t}} \cdot \left(z - a\right) \]

      16. lower--.f64 N/A

          \[\leadsto y - \frac{\color{blue}{y - x}}{t} \cdot \left(z - a\right) \]

      17. lower--.f64 71.7

          \[\leadsto y - \frac{y - x}{t} \cdot \color{blue}{\left(z - a\right)} \]

   10. Applied rewrites 71.7%

       \[\leadsto \color{blue}{y - \frac{y - x}{t} \cdot \left(z - a\right)} \]

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -9.99999999999999962e-292 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 1.9999999999999999e298

   1. Initial program 97.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   if -9.99999999999999962e-292 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 4.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 3.7

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 3.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z - t}{a}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z - t}{a}}, x\right) \]

      6. lower--.f64 1.6

         \[\leadsto \mathsf{fma}\left(y - x, \frac{\color{blue}{z - t}}{a}, x\right) \]

   7. Applied rewrites 1.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

   8. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   9. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(y - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{z \cdot \left(y - x\right)}{t}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      3. metadata-eval N/A

         \[\leadsto \left(y - \color{blue}{1} \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      4. *-lft-identity N/A

         \[\leadsto \left(y - \color{blue}{\frac{z \cdot \left(y - x\right)}{t}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      5. metadata-eval N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      6. *-lft-identity N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{\frac{a \cdot \left(y - x\right)}{t}} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{y - \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      8. div-sub N/A

         \[\leadsto y - \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      9. lower--.f64 N/A

         \[\leadsto \color{blue}{y - \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      10. div-sub N/A

          \[\leadsto y - \color{blue}{\left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      11. associate-/l* N/A

          \[\leadsto y - \left(\color{blue}{z \cdot \frac{y - x}{t}} - \frac{a \cdot \left(y - x\right)}{t}\right) \]

      12. associate-/l* N/A

          \[\leadsto y - \left(z \cdot \frac{y - x}{t} - \color{blue}{a \cdot \frac{y - x}{t}}\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t}} \cdot \left(z - a\right) \]

      16. lower--.f64 N/A

          \[\leadsto y - \frac{\color{blue}{y - x}}{t} \cdot \left(z - a\right) \]

      17. lower--.f64 82.9

          \[\leadsto y - \frac{y - x}{t} \cdot \color{blue}{\left(z - a\right)} \]

   10. Applied rewrites 82.9%

       \[\leadsto \color{blue}{y - \frac{y - x}{t} \cdot \left(z - a\right)} \]

   11. Taylor expanded in x around inf

       \[\leadsto y - -1 \cdot \color{blue}{\frac{x \cdot \left(z - a\right)}{t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 99.7%

       \[\leadsto y - \left(-x\right) \cdot \color{blue}{\frac{z - a}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 71.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)\\ \mathbf{if}\;a \leq -9 \cdot 10^{-80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-203}:\\ \;\;\;\;y - \frac{z \cdot \left(y - x\right)}{t}\\ \mathbf{elif}\;a \leq 1.95 \cdot 10^{-16}:\\ \;\;\;\;y - \left(-x\right) \cdot \frac{z - a}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y x) (/ (- z t) a) x)))
   (if (<= a -9e-80)
     t_1
     (if (<= a 5e-203)
       (- y (/ (* z (- y x)) t))
       (if (<= a 1.95e-16) (- y (* (- x) (/ (- z a) t))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - x), ((z - t) / a), x);
	double tmp;
	if (a <= -9e-80) {
		tmp = t_1;
	} else if (a <= 5e-203) {
		tmp = y - ((z * (y - x)) / t);
	} else if (a <= 1.95e-16) {
		tmp = y - (-x * ((z - a) / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - x), Float64(Float64(z - t) / a), x)
	tmp = 0.0
	if (a <= -9e-80)
		tmp = t_1;
	elseif (a <= 5e-203)
		tmp = Float64(y - Float64(Float64(z * Float64(y - x)) / t));
	elseif (a <= 1.95e-16)
		tmp = Float64(y - Float64(Float64(-x) * Float64(Float64(z - a) / t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -9e-80], t$95$1, If[LessEqual[a, 5e-203], N[(y - N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.95e-16], N[(y - N[((-x) * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.0000000000000006e-80 or 1.94999999999999989e-16 < a

   1. Initial program 71.4%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 65.2

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 65.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z - t}{a}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z - t}{a}}, x\right) \]

      6. lower--.f64 77.2

         \[\leadsto \mathsf{fma}\left(y - x, \frac{\color{blue}{z - t}}{a}, x\right) \]

   7. Applied rewrites 77.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

   if -9.0000000000000006e-80 < a < 5.0000000000000002e-203

   1. Initial program 67.8%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 53.7

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z - t}{a}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z - t}{a}}, x\right) \]

      6. lower--.f64 19.3

         \[\leadsto \mathsf{fma}\left(y - x, \frac{\color{blue}{z - t}}{a}, x\right) \]

   7. Applied rewrites 19.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

   8. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   9. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(y - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{z \cdot \left(y - x\right)}{t}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      3. metadata-eval N/A

         \[\leadsto \left(y - \color{blue}{1} \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      4. *-lft-identity N/A

         \[\leadsto \left(y - \color{blue}{\frac{z \cdot \left(y - x\right)}{t}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      5. metadata-eval N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      6. *-lft-identity N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{\frac{a \cdot \left(y - x\right)}{t}} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{y - \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      8. div-sub N/A

         \[\leadsto y - \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      9. lower--.f64 N/A

         \[\leadsto \color{blue}{y - \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      10. div-sub N/A

          \[\leadsto y - \color{blue}{\left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      11. associate-/l* N/A

          \[\leadsto y - \left(\color{blue}{z \cdot \frac{y - x}{t}} - \frac{a \cdot \left(y - x\right)}{t}\right) \]

      12. associate-/l* N/A

          \[\leadsto y - \left(z \cdot \frac{y - x}{t} - \color{blue}{a \cdot \frac{y - x}{t}}\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t}} \cdot \left(z - a\right) \]

      16. lower--.f64 N/A

          \[\leadsto y - \frac{\color{blue}{y - x}}{t} \cdot \left(z - a\right) \]

      17. lower--.f64 88.8

          \[\leadsto y - \frac{y - x}{t} \cdot \color{blue}{\left(z - a\right)} \]

   10. Applied rewrites 88.8%

       \[\leadsto \color{blue}{y - \frac{y - x}{t} \cdot \left(z - a\right)} \]

   11. Taylor expanded in z around inf

       \[\leadsto y - \frac{z \cdot \left(y - x\right)}{\color{blue}{t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 83.3%

       \[\leadsto y - \frac{z \cdot \left(y - x\right)}{\color{blue}{t}} \]

   if 5.0000000000000002e-203 < a < 1.94999999999999989e-16

   1. Initial program 52.6%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 43.2

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z - t}{a}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z - t}{a}}, x\right) \]

      6. lower--.f64 24.3

         \[\leadsto \mathsf{fma}\left(y - x, \frac{\color{blue}{z - t}}{a}, x\right) \]

   7. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

   8. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   9. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(y - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{z \cdot \left(y - x\right)}{t}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      3. metadata-eval N/A

         \[\leadsto \left(y - \color{blue}{1} \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      4. *-lft-identity N/A

         \[\leadsto \left(y - \color{blue}{\frac{z \cdot \left(y - x\right)}{t}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      5. metadata-eval N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      6. *-lft-identity N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{\frac{a \cdot \left(y - x\right)}{t}} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{y - \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      8. div-sub N/A

         \[\leadsto y - \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      9. lower--.f64 N/A

         \[\leadsto \color{blue}{y - \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      10. div-sub N/A

          \[\leadsto y - \color{blue}{\left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      11. associate-/l* N/A

          \[\leadsto y - \left(\color{blue}{z \cdot \frac{y - x}{t}} - \frac{a \cdot \left(y - x\right)}{t}\right) \]

      12. associate-/l* N/A

          \[\leadsto y - \left(z \cdot \frac{y - x}{t} - \color{blue}{a \cdot \frac{y - x}{t}}\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t}} \cdot \left(z - a\right) \]

      16. lower--.f64 N/A

          \[\leadsto y - \frac{\color{blue}{y - x}}{t} \cdot \left(z - a\right) \]

      17. lower--.f64 77.7

          \[\leadsto y - \frac{y - x}{t} \cdot \color{blue}{\left(z - a\right)} \]

   10. Applied rewrites 77.7%

       \[\leadsto \color{blue}{y - \frac{y - x}{t} \cdot \left(z - a\right)} \]

   11. Taylor expanded in x around inf

       \[\leadsto y - -1 \cdot \color{blue}{\frac{x \cdot \left(z - a\right)}{t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 74.7%

       \[\leadsto y - \left(-x\right) \cdot \color{blue}{\frac{z - a}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 60.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{if}\;a \leq -1.1 \cdot 10^{-76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-304}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{a - t}\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{z - t}{t} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z a) (- y x) x)))
   (if (<= a -1.1e-76)
     t_1
     (if (<= a 1.3e-304)
       (* (- y x) (/ z (- a t)))
       (if (<= a 8.5e-40) (* (/ (- z t) t) (- y)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z / a), (y - x), x);
	double tmp;
	if (a <= -1.1e-76) {
		tmp = t_1;
	} else if (a <= 1.3e-304) {
		tmp = (y - x) * (z / (a - t));
	} else if (a <= 8.5e-40) {
		tmp = ((z - t) / t) * -y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z / a), Float64(y - x), x)
	tmp = 0.0
	if (a <= -1.1e-76)
		tmp = t_1;
	elseif (a <= 1.3e-304)
		tmp = Float64(Float64(y - x) * Float64(z / Float64(a - t)));
	elseif (a <= 8.5e-40)
		tmp = Float64(Float64(Float64(z - t) / t) * Float64(-y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.1e-76], t$95$1, If[LessEqual[a, 1.3e-304], N[(N[(y - x), $MachinePrecision] * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 8.5e-40], N[(N[(N[(z - t), $MachinePrecision] / t), $MachinePrecision] * (-y)), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.1e-76 or 8.4999999999999998e-40 < a

   1. Initial program 70.6%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(y - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{a}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y - x}{a} \cdot z} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - x}{a}}, z, x\right) \]

      6. lower--.f64 67.3

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - x}}{a}, z, x\right) \]

   5. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 68.9%

      \[\leadsto \mathsf{fma}\left(\frac{z}{a}, \color{blue}{y - x}, x\right) \]

   if -1.1e-76 < a < 1.29999999999999998e-304

   1. Initial program 68.3%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{a - t}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot z}}{a - t} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{a - t}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{a - t}} \]

      6. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(y - x\right)} \cdot \frac{z}{a - t} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(y - x\right) \cdot \color{blue}{\frac{z}{a - t}} \]

      8. lower--.f64 69.3

         \[\leadsto \left(y - x\right) \cdot \frac{z}{\color{blue}{a - t}} \]

   5. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{a - t}} \]

   if 1.29999999999999998e-304 < a < 8.4999999999999998e-40

   1. Initial program 60.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 51.7

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{t}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{t} + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\left(y - x\right) \cdot \left(z - t\right)}{t}\right)\right)} + x \]

      3. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - t}{t}}\right)\right) + x \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - t}{t}} + x \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - t}{t} + x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - t}{t}, x\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(y - x\right)\right)}, \frac{z - t}{t}, x\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-\left(y - x\right)}, \frac{z - t}{t}, x\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(-\color{blue}{\left(y - x\right)}, \frac{z - t}{t}, x\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-\left(y - x\right), \color{blue}{\frac{z - t}{t}}, x\right) \]

      11. lower--.f64 58.7

          \[\leadsto \mathsf{fma}\left(-\left(y - x\right), \frac{\color{blue}{z - t}}{t}, x\right) \]

   7. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\left(y - x\right), \frac{z - t}{t}, x\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{y \cdot \left(z - t\right)}{t}} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.4%

       \[\leadsto \frac{z - t}{t} \cdot \color{blue}{\left(-y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-304}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{a - t}\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{z - t}{t} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{if}\;a \leq -1.35 \cdot 10^{-97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-304}:\\ \;\;\;\;\frac{x - y}{t} \cdot z\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{z - t}{t} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z a) (- y x) x)))
   (if (<= a -1.35e-97)
     t_1
     (if (<= a 1.1e-304)
       (* (/ (- x y) t) z)
       (if (<= a 8.5e-40) (* (/ (- z t) t) (- y)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z / a), (y - x), x);
	double tmp;
	if (a <= -1.35e-97) {
		tmp = t_1;
	} else if (a <= 1.1e-304) {
		tmp = ((x - y) / t) * z;
	} else if (a <= 8.5e-40) {
		tmp = ((z - t) / t) * -y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z / a), Float64(y - x), x)
	tmp = 0.0
	if (a <= -1.35e-97)
		tmp = t_1;
	elseif (a <= 1.1e-304)
		tmp = Float64(Float64(Float64(x - y) / t) * z);
	elseif (a <= 8.5e-40)
		tmp = Float64(Float64(Float64(z - t) / t) * Float64(-y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.35e-97], t$95$1, If[LessEqual[a, 1.1e-304], N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[a, 8.5e-40], N[(N[(N[(z - t), $MachinePrecision] / t), $MachinePrecision] * (-y)), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.34999999999999993e-97 or 8.4999999999999998e-40 < a

   1. Initial program 69.4%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(y - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{a}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y - x}{a} \cdot z} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - x}{a}}, z, x\right) \]

      6. lower--.f64 65.1

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - x}}{a}, z, x\right) \]

   5. Applied rewrites 65.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.5%

      \[\leadsto \mathsf{fma}\left(\frac{z}{a}, \color{blue}{y - x}, x\right) \]

   if -1.34999999999999993e-97 < a < 1.1e-304

   1. Initial program 72.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 49.6

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 49.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z - t}{a}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z - t}{a}}, x\right) \]

      6. lower--.f64 18.6

         \[\leadsto \mathsf{fma}\left(y - x, \frac{\color{blue}{z - t}}{a}, x\right) \]

   7. Applied rewrites 18.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

   8. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   9. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(y - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{z \cdot \left(y - x\right)}{t}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      3. metadata-eval N/A

         \[\leadsto \left(y - \color{blue}{1} \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      4. *-lft-identity N/A

         \[\leadsto \left(y - \color{blue}{\frac{z \cdot \left(y - x\right)}{t}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      5. metadata-eval N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      6. *-lft-identity N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{\frac{a \cdot \left(y - x\right)}{t}} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{y - \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      8. div-sub N/A

         \[\leadsto y - \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      9. lower--.f64 N/A

         \[\leadsto \color{blue}{y - \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      10. div-sub N/A

          \[\leadsto y - \color{blue}{\left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      11. associate-/l* N/A

          \[\leadsto y - \left(\color{blue}{z \cdot \frac{y - x}{t}} - \frac{a \cdot \left(y - x\right)}{t}\right) \]

      12. associate-/l* N/A

          \[\leadsto y - \left(z \cdot \frac{y - x}{t} - \color{blue}{a \cdot \frac{y - x}{t}}\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t}} \cdot \left(z - a\right) \]

      16. lower--.f64 N/A

          \[\leadsto y - \frac{\color{blue}{y - x}}{t} \cdot \left(z - a\right) \]

      17. lower--.f64 89.4

          \[\leadsto y - \frac{y - x}{t} \cdot \color{blue}{\left(z - a\right)} \]

   10. Applied rewrites 89.4%

       \[\leadsto \color{blue}{y - \frac{y - x}{t} \cdot \left(z - a\right)} \]

   11. Taylor expanded in z around inf

       \[\leadsto z \cdot \color{blue}{\left(\frac{x}{t} - \frac{y}{t}\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 70.2%

       \[\leadsto \frac{x - y}{t} \cdot \color{blue}{z} \]

   if 1.1e-304 < a < 8.4999999999999998e-40

   1. Initial program 60.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 51.7

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{t}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{t} + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\left(y - x\right) \cdot \left(z - t\right)}{t}\right)\right)} + x \]

      3. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - t}{t}}\right)\right) + x \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - t}{t}} + x \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - t}{t} + x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - t}{t}, x\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(y - x\right)\right)}, \frac{z - t}{t}, x\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-\left(y - x\right)}, \frac{z - t}{t}, x\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(-\color{blue}{\left(y - x\right)}, \frac{z - t}{t}, x\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-\left(y - x\right), \color{blue}{\frac{z - t}{t}}, x\right) \]

      11. lower--.f64 58.7

          \[\leadsto \mathsf{fma}\left(-\left(y - x\right), \frac{\color{blue}{z - t}}{t}, x\right) \]

   7. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\left(y - x\right), \frac{z - t}{t}, x\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{y \cdot \left(z - t\right)}{t}} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.4%

       \[\leadsto \frac{z - t}{t} \cdot \color{blue}{\left(-y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-97}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-304}:\\ \;\;\;\;\frac{x - y}{t} \cdot z\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{z - t}{t} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.3 \cdot 10^{-79} \lor \neg \left(a \leq 2.6 \cdot 10^{+34}\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{y - x}{t} \cdot \left(z - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -5.3e-79) (not (<= a 2.6e+34)))
   (fma (- y x) (/ (- z t) a) x)
   (- y (* (/ (- y x) t) (- z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -5.3e-79) || !(a <= 2.6e+34)) {
		tmp = fma((y - x), ((z - t) / a), x);
	} else {
		tmp = y - (((y - x) / t) * (z - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -5.3e-79) || !(a <= 2.6e+34))
		tmp = fma(Float64(y - x), Float64(Float64(z - t) / a), x);
	else
		tmp = Float64(y - Float64(Float64(Float64(y - x) / t) * Float64(z - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -5.3e-79], N[Not[LessEqual[a, 2.6e+34]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(y - N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.2999999999999998e-79 or 2.59999999999999997e34 < a

   1. Initial program 70.9%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 64.4

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 64.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z - t}{a}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z - t}{a}}, x\right) \]

      6. lower--.f64 77.5

         \[\leadsto \mathsf{fma}\left(y - x, \frac{\color{blue}{z - t}}{a}, x\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

   if -5.2999999999999998e-79 < a < 2.59999999999999997e34

   1. Initial program 64.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 52.2

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z - t}{a}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z - t}{a}}, x\right) \]

      6. lower--.f64 23.3

         \[\leadsto \mathsf{fma}\left(y - x, \frac{\color{blue}{z - t}}{a}, x\right) \]

   7. Applied rewrites 23.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

   8. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   9. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(y - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{z \cdot \left(y - x\right)}{t}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      3. metadata-eval N/A

         \[\leadsto \left(y - \color{blue}{1} \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      4. *-lft-identity N/A

         \[\leadsto \left(y - \color{blue}{\frac{z \cdot \left(y - x\right)}{t}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      5. metadata-eval N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      6. *-lft-identity N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{\frac{a \cdot \left(y - x\right)}{t}} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{y - \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      8. div-sub N/A

         \[\leadsto y - \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      9. lower--.f64 N/A

         \[\leadsto \color{blue}{y - \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      10. div-sub N/A

          \[\leadsto y - \color{blue}{\left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      11. associate-/l* N/A

          \[\leadsto y - \left(\color{blue}{z \cdot \frac{y - x}{t}} - \frac{a \cdot \left(y - x\right)}{t}\right) \]

      12. associate-/l* N/A

          \[\leadsto y - \left(z \cdot \frac{y - x}{t} - \color{blue}{a \cdot \frac{y - x}{t}}\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t}} \cdot \left(z - a\right) \]

      16. lower--.f64 N/A

          \[\leadsto y - \frac{\color{blue}{y - x}}{t} \cdot \left(z - a\right) \]

      17. lower--.f64 84.8

          \[\leadsto y - \frac{y - x}{t} \cdot \color{blue}{\left(z - a\right)} \]

   10. Applied rewrites 84.8%

       \[\leadsto \color{blue}{y - \frac{y - x}{t} \cdot \left(z - a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.3 \cdot 10^{-79} \lor \neg \left(a \leq 2.6 \cdot 10^{+34}\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{y - x}{t} \cdot \left(z - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-80} \lor \neg \left(a \leq 3.3 \cdot 10^{-29}\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{z \cdot \left(y - x\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -9e-80) (not (<= a 3.3e-29)))
   (fma (- y x) (/ (- z t) a) x)
   (- y (/ (* z (- y x)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -9e-80) || !(a <= 3.3e-29)) {
		tmp = fma((y - x), ((z - t) / a), x);
	} else {
		tmp = y - ((z * (y - x)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -9e-80) || !(a <= 3.3e-29))
		tmp = fma(Float64(y - x), Float64(Float64(z - t) / a), x);
	else
		tmp = Float64(y - Float64(Float64(z * Float64(y - x)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -9e-80], N[Not[LessEqual[a, 3.3e-29]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(y - N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.0000000000000006e-80 or 3.30000000000000028e-29 < a

   1. Initial program 71.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 65.5

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z - t}{a}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z - t}{a}}, x\right) \]

      6. lower--.f64 76.7

         \[\leadsto \mathsf{fma}\left(y - x, \frac{\color{blue}{z - t}}{a}, x\right) \]

   7. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

   if -9.0000000000000006e-80 < a < 3.30000000000000028e-29

   1. Initial program 63.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 50.0

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z - t}{a}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z - t}{a}}, x\right) \]

      6. lower--.f64 20.9

         \[\leadsto \mathsf{fma}\left(y - x, \frac{\color{blue}{z - t}}{a}, x\right) \]

   7. Applied rewrites 20.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

   8. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   9. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(y - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{z \cdot \left(y - x\right)}{t}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      3. metadata-eval N/A

         \[\leadsto \left(y - \color{blue}{1} \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      4. *-lft-identity N/A

         \[\leadsto \left(y - \color{blue}{\frac{z \cdot \left(y - x\right)}{t}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      5. metadata-eval N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      6. *-lft-identity N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{\frac{a \cdot \left(y - x\right)}{t}} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{y - \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      8. div-sub N/A

         \[\leadsto y - \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      9. lower--.f64 N/A

         \[\leadsto \color{blue}{y - \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      10. div-sub N/A

          \[\leadsto y - \color{blue}{\left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      11. associate-/l* N/A

          \[\leadsto y - \left(\color{blue}{z \cdot \frac{y - x}{t}} - \frac{a \cdot \left(y - x\right)}{t}\right) \]

      12. associate-/l* N/A

          \[\leadsto y - \left(z \cdot \frac{y - x}{t} - \color{blue}{a \cdot \frac{y - x}{t}}\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t}} \cdot \left(z - a\right) \]

      16. lower--.f64 N/A

          \[\leadsto y - \frac{\color{blue}{y - x}}{t} \cdot \left(z - a\right) \]

      17. lower--.f64 85.2

          \[\leadsto y - \frac{y - x}{t} \cdot \color{blue}{\left(z - a\right)} \]

   10. Applied rewrites 85.2%

       \[\leadsto \color{blue}{y - \frac{y - x}{t} \cdot \left(z - a\right)} \]

   11. Taylor expanded in z around inf

       \[\leadsto y - \frac{z \cdot \left(y - x\right)}{\color{blue}{t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 76.9%

       \[\leadsto y - \frac{z \cdot \left(y - x\right)}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-80} \lor \neg \left(a \leq 3.3 \cdot 10^{-29}\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{z \cdot \left(y - x\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-78} \lor \neg \left(a \leq 5.1 \cdot 10^{-39}\right):\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{z \cdot \left(y - x\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1e-78) (not (<= a 5.1e-39)))
   (fma (- z t) (/ (- y x) a) x)
   (- y (/ (* z (- y x)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1e-78) || !(a <= 5.1e-39)) {
		tmp = fma((z - t), ((y - x) / a), x);
	} else {
		tmp = y - ((z * (y - x)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1e-78) || !(a <= 5.1e-39))
		tmp = fma(Float64(z - t), Float64(Float64(y - x) / a), x);
	else
		tmp = Float64(y - Float64(Float64(z * Float64(y - x)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1e-78], N[Not[LessEqual[a, 5.1e-39]], $MachinePrecision]], N[(N[(z - t), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(y - N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.99999999999999999e-79 or 5.09999999999999988e-39 < a

   1. Initial program 70.8%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z - t\right) \cdot \left(y - x\right)}}{a} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(z - t\right) \cdot \frac{y - x}{a}} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z - t}, \frac{y - x}{a}, x\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(z - t, \color{blue}{\frac{y - x}{a}}, x\right) \]

      7. lower--.f64 75.1

         \[\leadsto \mathsf{fma}\left(z - t, \frac{\color{blue}{y - x}}{a}, x\right) \]

   5. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)} \]

   if -9.99999999999999999e-79 < a < 5.09999999999999988e-39

   1. Initial program 63.9%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 49.6

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 49.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z - t}{a}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z - t}{a}}, x\right) \]

      6. lower--.f64 21.7

         \[\leadsto \mathsf{fma}\left(y - x, \frac{\color{blue}{z - t}}{a}, x\right) \]

   7. Applied rewrites 21.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

   8. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   9. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(y - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{z \cdot \left(y - x\right)}{t}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      3. metadata-eval N/A

         \[\leadsto \left(y - \color{blue}{1} \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      4. *-lft-identity N/A

         \[\leadsto \left(y - \color{blue}{\frac{z \cdot \left(y - x\right)}{t}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      5. metadata-eval N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      6. *-lft-identity N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{\frac{a \cdot \left(y - x\right)}{t}} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{y - \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      8. div-sub N/A

         \[\leadsto y - \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      9. lower--.f64 N/A

         \[\leadsto \color{blue}{y - \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      10. div-sub N/A

          \[\leadsto y - \color{blue}{\left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      11. associate-/l* N/A

          \[\leadsto y - \left(\color{blue}{z \cdot \frac{y - x}{t}} - \frac{a \cdot \left(y - x\right)}{t}\right) \]

      12. associate-/l* N/A

          \[\leadsto y - \left(z \cdot \frac{y - x}{t} - \color{blue}{a \cdot \frac{y - x}{t}}\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t}} \cdot \left(z - a\right) \]

      16. lower--.f64 N/A

          \[\leadsto y - \frac{\color{blue}{y - x}}{t} \cdot \left(z - a\right) \]

      17. lower--.f64 84.5

          \[\leadsto y - \frac{y - x}{t} \cdot \color{blue}{\left(z - a\right)} \]

   10. Applied rewrites 84.5%

       \[\leadsto \color{blue}{y - \frac{y - x}{t} \cdot \left(z - a\right)} \]

   11. Taylor expanded in z around inf

       \[\leadsto y - \frac{z \cdot \left(y - x\right)}{\color{blue}{t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 76.2%

       \[\leadsto y - \frac{z \cdot \left(y - x\right)}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-78} \lor \neg \left(a \leq 5.1 \cdot 10^{-39}\right):\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{z \cdot \left(y - x\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{-79} \lor \neg \left(a \leq 3.3 \cdot 10^{-29}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{z \cdot \left(y - x\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -2.9e-79) (not (<= a 3.3e-29)))
   (fma (/ z a) (- y x) x)
   (- y (/ (* z (- y x)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2.9e-79) || !(a <= 3.3e-29)) {
		tmp = fma((z / a), (y - x), x);
	} else {
		tmp = y - ((z * (y - x)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -2.9e-79) || !(a <= 3.3e-29))
		tmp = fma(Float64(z / a), Float64(y - x), x);
	else
		tmp = Float64(y - Float64(Float64(z * Float64(y - x)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -2.9e-79], N[Not[LessEqual[a, 3.3e-29]], $MachinePrecision]], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(y - N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.9000000000000001e-79 or 3.30000000000000028e-29 < a

   1. Initial program 71.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(y - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{a}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y - x}{a} \cdot z} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - x}{a}}, z, x\right) \]

      6. lower--.f64 67.1

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - x}}{a}, z, x\right) \]

   5. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.3%

      \[\leadsto \mathsf{fma}\left(\frac{z}{a}, \color{blue}{y - x}, x\right) \]

   if -2.9000000000000001e-79 < a < 3.30000000000000028e-29

   1. Initial program 63.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 50.0

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z - t}{a}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z - t}{a}}, x\right) \]

      6. lower--.f64 20.9

         \[\leadsto \mathsf{fma}\left(y - x, \frac{\color{blue}{z - t}}{a}, x\right) \]

   7. Applied rewrites 20.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

   8. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   9. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(y - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{z \cdot \left(y - x\right)}{t}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      3. metadata-eval N/A

         \[\leadsto \left(y - \color{blue}{1} \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      4. *-lft-identity N/A

         \[\leadsto \left(y - \color{blue}{\frac{z \cdot \left(y - x\right)}{t}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      5. metadata-eval N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      6. *-lft-identity N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{\frac{a \cdot \left(y - x\right)}{t}} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{y - \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      8. div-sub N/A

         \[\leadsto y - \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      9. lower--.f64 N/A

         \[\leadsto \color{blue}{y - \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      10. div-sub N/A

          \[\leadsto y - \color{blue}{\left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      11. associate-/l* N/A

          \[\leadsto y - \left(\color{blue}{z \cdot \frac{y - x}{t}} - \frac{a \cdot \left(y - x\right)}{t}\right) \]

      12. associate-/l* N/A

          \[\leadsto y - \left(z \cdot \frac{y - x}{t} - \color{blue}{a \cdot \frac{y - x}{t}}\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t}} \cdot \left(z - a\right) \]

      16. lower--.f64 N/A

          \[\leadsto y - \frac{\color{blue}{y - x}}{t} \cdot \left(z - a\right) \]

      17. lower--.f64 85.2

          \[\leadsto y - \frac{y - x}{t} \cdot \color{blue}{\left(z - a\right)} \]

   10. Applied rewrites 85.2%

       \[\leadsto \color{blue}{y - \frac{y - x}{t} \cdot \left(z - a\right)} \]

   11. Taylor expanded in z around inf

       \[\leadsto y - \frac{z \cdot \left(y - x\right)}{\color{blue}{t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 76.9%

       \[\leadsto y - \frac{z \cdot \left(y - x\right)}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{-79} \lor \neg \left(a \leq 3.3 \cdot 10^{-29}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{z \cdot \left(y - x\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+178} \lor \neg \left(t \leq 1.95 \cdot 10^{+112}\right):\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -8e+178) (not (<= t 1.95e+112)))
   (fma a (/ (- y x) t) y)
   (fma (/ z a) (- y x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -8e+178) || !(t <= 1.95e+112)) {
		tmp = fma(a, ((y - x) / t), y);
	} else {
		tmp = fma((z / a), (y - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -8e+178) || !(t <= 1.95e+112))
		tmp = fma(a, Float64(Float64(y - x) / t), y);
	else
		tmp = fma(Float64(z / a), Float64(y - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -8e+178], N[Not[LessEqual[t, 1.95e+112]], $MachinePrecision]], N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8.0000000000000004e178 or 1.94999999999999984e112 < t

   1. Initial program 30.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 15.7

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 15.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z - t}{a}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z - t}{a}}, x\right) \]

      6. lower--.f64 10.5

         \[\leadsto \mathsf{fma}\left(y - x, \frac{\color{blue}{z - t}}{a}, x\right) \]

   7. Applied rewrites 10.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

   8. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   9. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(y - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{z \cdot \left(y - x\right)}{t}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      3. metadata-eval N/A

         \[\leadsto \left(y - \color{blue}{1} \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      4. *-lft-identity N/A

         \[\leadsto \left(y - \color{blue}{\frac{z \cdot \left(y - x\right)}{t}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      5. metadata-eval N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      6. *-lft-identity N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{\frac{a \cdot \left(y - x\right)}{t}} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{y - \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      8. div-sub N/A

         \[\leadsto y - \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      9. lower--.f64 N/A

         \[\leadsto \color{blue}{y - \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      10. div-sub N/A

          \[\leadsto y - \color{blue}{\left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      11. associate-/l* N/A

          \[\leadsto y - \left(\color{blue}{z \cdot \frac{y - x}{t}} - \frac{a \cdot \left(y - x\right)}{t}\right) \]

      12. associate-/l* N/A

          \[\leadsto y - \left(z \cdot \frac{y - x}{t} - \color{blue}{a \cdot \frac{y - x}{t}}\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t}} \cdot \left(z - a\right) \]

      16. lower--.f64 N/A

          \[\leadsto y - \frac{\color{blue}{y - x}}{t} \cdot \left(z - a\right) \]

      17. lower--.f64 87.2

          \[\leadsto y - \frac{y - x}{t} \cdot \color{blue}{\left(z - a\right)} \]

   10. Applied rewrites 87.2%

       \[\leadsto \color{blue}{y - \frac{y - x}{t} \cdot \left(z - a\right)} \]

   11. Taylor expanded in z around 0

       \[\leadsto y - \color{blue}{-1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 64.8%

       \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{y - x}{t}}, y\right) \]

   if -8.0000000000000004e178 < t < 1.94999999999999984e112

   1. Initial program 80.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(y - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{a}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y - x}{a} \cdot z} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - x}{a}}, z, x\right) \]

      6. lower--.f64 61.4

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - x}}{a}, z, x\right) \]

   5. Applied rewrites 61.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 61.6%

      \[\leadsto \mathsf{fma}\left(\frac{z}{a}, \color{blue}{y - x}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+178} \lor \neg \left(t \leq 1.95 \cdot 10^{+112}\right):\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-97} \lor \neg \left(a \leq 2.9 \cdot 10^{-18}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.35e-97) (not (<= a 2.9e-18)))
   (fma (/ z a) (- y x) x)
   (* (/ (- x y) t) z)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.35e-97) || !(a <= 2.9e-18)) {
		tmp = fma((z / a), (y - x), x);
	} else {
		tmp = ((x - y) / t) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.35e-97) || !(a <= 2.9e-18))
		tmp = fma(Float64(z / a), Float64(y - x), x);
	else
		tmp = Float64(Float64(Float64(x - y) / t) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.35e-97], N[Not[LessEqual[a, 2.9e-18]], $MachinePrecision]], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.34999999999999993e-97 or 2.9e-18 < a

   1. Initial program 69.8%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(y - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{a}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y - x}{a} \cdot z} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - x}{a}}, z, x\right) \]

      6. lower--.f64 65.4

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - x}}{a}, z, x\right) \]

   5. Applied rewrites 65.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.8%

      \[\leadsto \mathsf{fma}\left(\frac{z}{a}, \color{blue}{y - x}, x\right) \]

   if -1.34999999999999993e-97 < a < 2.9e-18

   1. Initial program 64.9%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 51.3

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 51.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z - t}{a}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z - t}{a}}, x\right) \]

      6. lower--.f64 20.3

         \[\leadsto \mathsf{fma}\left(y - x, \frac{\color{blue}{z - t}}{a}, x\right) \]

   7. Applied rewrites 20.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

   8. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   9. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(y - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{z \cdot \left(y - x\right)}{t}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      3. metadata-eval N/A

         \[\leadsto \left(y - \color{blue}{1} \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      4. *-lft-identity N/A

         \[\leadsto \left(y - \color{blue}{\frac{z \cdot \left(y - x\right)}{t}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      5. metadata-eval N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      6. *-lft-identity N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{\frac{a \cdot \left(y - x\right)}{t}} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{y - \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      8. div-sub N/A

         \[\leadsto y - \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      9. lower--.f64 N/A

         \[\leadsto \color{blue}{y - \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      10. div-sub N/A

          \[\leadsto y - \color{blue}{\left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      11. associate-/l* N/A

          \[\leadsto y - \left(\color{blue}{z \cdot \frac{y - x}{t}} - \frac{a \cdot \left(y - x\right)}{t}\right) \]

      12. associate-/l* N/A

          \[\leadsto y - \left(z \cdot \frac{y - x}{t} - \color{blue}{a \cdot \frac{y - x}{t}}\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t}} \cdot \left(z - a\right) \]

      16. lower--.f64 N/A

          \[\leadsto y - \frac{\color{blue}{y - x}}{t} \cdot \left(z - a\right) \]

      17. lower--.f64 87.1

          \[\leadsto y - \frac{y - x}{t} \cdot \color{blue}{\left(z - a\right)} \]

   10. Applied rewrites 87.1%

       \[\leadsto \color{blue}{y - \frac{y - x}{t} \cdot \left(z - a\right)} \]

   11. Taylor expanded in z around inf

       \[\leadsto z \cdot \color{blue}{\left(\frac{x}{t} - \frac{y}{t}\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 54.0%

       \[\leadsto \frac{x - y}{t} \cdot \color{blue}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-97} \lor \neg \left(a \leq 2.9 \cdot 10^{-18}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{t} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.46 \cdot 10^{-97} \lor \neg \left(a \leq 2.55 \cdot 10^{+34}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.46e-97) (not (<= a 2.55e+34)))
   (fma (/ y a) z x)
   (* (/ (- x y) t) z)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.46e-97) || !(a <= 2.55e+34)) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = ((x - y) / t) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.46e-97) || !(a <= 2.55e+34))
		tmp = fma(Float64(y / a), z, x);
	else
		tmp = Float64(Float64(Float64(x - y) / t) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.46e-97], N[Not[LessEqual[a, 2.55e+34]], $MachinePrecision]], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.4600000000000001e-97 or 2.55000000000000018e34 < a

   1. Initial program 69.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(y - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{a}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y - x}{a} \cdot z} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - x}{a}}, z, x\right) \]

      6. lower--.f64 65.3

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - x}}{a}, z, x\right) \]

   5. Applied rewrites 65.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{y}{a}, z, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 58.5%

      \[\leadsto \mathsf{fma}\left(\frac{y}{a}, z, x\right) \]

   if -1.4600000000000001e-97 < a < 2.55000000000000018e34

   1. Initial program 66.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 53.1

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 53.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z - t}{a}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z - t}{a}}, x\right) \]

      6. lower--.f64 23.1

         \[\leadsto \mathsf{fma}\left(y - x, \frac{\color{blue}{z - t}}{a}, x\right) \]

   7. Applied rewrites 23.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)} \]

   8. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   9. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(y - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{z \cdot \left(y - x\right)}{t}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      3. metadata-eval N/A

         \[\leadsto \left(y - \color{blue}{1} \cdot \frac{z \cdot \left(y - x\right)}{t}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      4. *-lft-identity N/A

         \[\leadsto \left(y - \color{blue}{\frac{z \cdot \left(y - x\right)}{t}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      5. metadata-eval N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(y - x\right)}{t} \]

      6. *-lft-identity N/A

         \[\leadsto \left(y - \frac{z \cdot \left(y - x\right)}{t}\right) + \color{blue}{\frac{a \cdot \left(y - x\right)}{t}} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{y - \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      8. div-sub N/A

         \[\leadsto y - \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      9. lower--.f64 N/A

         \[\leadsto \color{blue}{y - \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      10. div-sub N/A

          \[\leadsto y - \color{blue}{\left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      11. associate-/l* N/A

          \[\leadsto y - \left(\color{blue}{z \cdot \frac{y - x}{t}} - \frac{a \cdot \left(y - x\right)}{t}\right) \]

      12. associate-/l* N/A

          \[\leadsto y - \left(z \cdot \frac{y - x}{t} - \color{blue}{a \cdot \frac{y - x}{t}}\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto y - \color{blue}{\frac{y - x}{t}} \cdot \left(z - a\right) \]

      16. lower--.f64 N/A

          \[\leadsto y - \frac{\color{blue}{y - x}}{t} \cdot \left(z - a\right) \]

      17. lower--.f64 86.5

          \[\leadsto y - \frac{y - x}{t} \cdot \color{blue}{\left(z - a\right)} \]

   10. Applied rewrites 86.5%

       \[\leadsto \color{blue}{y - \frac{y - x}{t} \cdot \left(z - a\right)} \]

   11. Taylor expanded in z around inf

       \[\leadsto z \cdot \color{blue}{\left(\frac{x}{t} - \frac{y}{t}\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 54.3%

       \[\leadsto \frac{x - y}{t} \cdot \color{blue}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.46 \cdot 10^{-97} \lor \neg \left(a \leq 2.55 \cdot 10^{+34}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{t} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.46 \cdot 10^{-97} \lor \neg \left(a \leq 1.52 \cdot 10^{+34}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.46e-97) (not (<= a 1.52e+34)))
   (fma (/ y a) z x)
   (/ (* (- x y) z) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.46e-97) || !(a <= 1.52e+34)) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = ((x - y) * z) / t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.46e-97) || !(a <= 1.52e+34))
		tmp = fma(Float64(y / a), z, x);
	else
		tmp = Float64(Float64(Float64(x - y) * z) / t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.46e-97], N[Not[LessEqual[a, 1.52e+34]], $MachinePrecision]], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(N[(x - y), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.4600000000000001e-97 or 1.52000000000000001e34 < a

   1. Initial program 69.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(y - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{a}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y - x}{a} \cdot z} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - x}{a}}, z, x\right) \]

      6. lower--.f64 65.3

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - x}}{a}, z, x\right) \]

   5. Applied rewrites 65.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{y}{a}, z, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 58.5%

      \[\leadsto \mathsf{fma}\left(\frac{y}{a}, z, x\right) \]

   if -1.4600000000000001e-97 < a < 1.52000000000000001e34

   1. Initial program 66.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 53.1

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 53.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{t}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{t} + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\left(y - x\right) \cdot \left(z - t\right)}{t}\right)\right)} + x \]

      3. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - t}{t}}\right)\right) + x \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - t}{t}} + x \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - t}{t} + x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - t}{t}, x\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(y - x\right)\right)}, \frac{z - t}{t}, x\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-\left(y - x\right)}, \frac{z - t}{t}, x\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(-\color{blue}{\left(y - x\right)}, \frac{z - t}{t}, x\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-\left(y - x\right), \color{blue}{\frac{z - t}{t}}, x\right) \]

      11. lower--.f64 64.7

          \[\leadsto \mathsf{fma}\left(-\left(y - x\right), \frac{\color{blue}{z - t}}{t}, x\right) \]

   7. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\left(y - x\right), \frac{z - t}{t}, x\right)} \]

   8. Taylor expanded in z around inf

      \[\leadsto z \cdot \color{blue}{\left(\frac{x}{t} - \frac{y}{t}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 50.5%

       \[\leadsto \frac{\left(x - y\right) \cdot z}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.46 \cdot 10^{-97} \lor \neg \left(a \leq 1.52 \cdot 10^{+34}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+178} \lor \neg \left(t \leq 1.9 \cdot 10^{+132}\right):\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -7.5e+178) (not (<= t 1.9e+132))) y (fma (/ y a) z x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -7.5e+178) || !(t <= 1.9e+132)) {
		tmp = y;
	} else {
		tmp = fma((y / a), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -7.5e+178) || !(t <= 1.9e+132))
		tmp = y;
	else
		tmp = fma(Float64(y / a), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -7.5e+178], N[Not[LessEqual[t, 1.9e+132]], $MachinePrecision]], y, N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.4999999999999995e178 or 1.90000000000000003e132 < t

   1. Initial program 28.9%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 11.7

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 11.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{t}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{t} + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\left(y - x\right) \cdot \left(z - t\right)}{t}\right)\right)} + x \]

      3. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - t}{t}}\right)\right) + x \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - t}{t}} + x \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - t}{t} + x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - t}{t}, x\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(y - x\right)\right)}, \frac{z - t}{t}, x\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-\left(y - x\right)}, \frac{z - t}{t}, x\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(-\color{blue}{\left(y - x\right)}, \frac{z - t}{t}, x\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-\left(y - x\right), \color{blue}{\frac{z - t}{t}}, x\right) \]

      11. lower--.f64 45.1

          \[\leadsto \mathsf{fma}\left(-\left(y - x\right), \frac{\color{blue}{z - t}}{t}, x\right) \]

   7. Applied rewrites 45.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\left(y - x\right), \frac{z - t}{t}, x\right)} \]

   8. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{-1 \cdot \left(x - y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 35.1%

       \[\leadsto x - \color{blue}{\left(x - y\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 55.2%

       \[\leadsto y + 0 \]

   if -7.4999999999999995e178 < t < 1.90000000000000003e132

   1. Initial program 79.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(y - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{a}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y - x}{a} \cdot z} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - x}{a}}, z, x\right) \]

      6. lower--.f64 60.9

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - x}}{a}, z, x\right) \]

   5. Applied rewrites 60.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{y}{a}, z, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 53.4%

      \[\leadsto \mathsf{fma}\left(\frac{y}{a}, z, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+178} \lor \neg \left(t \leq 1.9 \cdot 10^{+132}\right):\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 39.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+106} \lor \neg \left(z \leq 1.12 \cdot 10^{+83}\right):\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.6e+106) (not (<= z 1.12e+83))) (* y (/ z a)) (- x (- y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.6e+106) || !(z <= 1.12e+83)) {
		tmp = y * (z / a);
	} else {
		tmp = x - -y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-1.6d+106)) .or. (.not. (z <= 1.12d+83))) then
        tmp = y * (z / a)
    else
        tmp = x - -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.6e+106) || !(z <= 1.12e+83)) {
		tmp = y * (z / a);
	} else {
		tmp = x - -y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.6e+106) or not (z <= 1.12e+83):
		tmp = y * (z / a)
	else:
		tmp = x - -y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.6e+106) || !(z <= 1.12e+83))
		tmp = Float64(y * Float64(z / a));
	else
		tmp = Float64(x - Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -1.6e+106) || ~((z <= 1.12e+83)))
		tmp = y * (z / a);
	else
		tmp = x - -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.6e+106], N[Not[LessEqual[z, 1.12e+83]], $MachinePrecision]], N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision], N[(x - (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5999999999999999e106 or 1.12e83 < z

   1. Initial program 72.9%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(y - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{a}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y - x}{a} \cdot z} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - x}{a}}, z, x\right) \]

      6. lower--.f64 55.8

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - x}}{a}, z, x\right) \]

   5. Applied rewrites 55.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{y \cdot z}{\color{blue}{a}} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.6%

      \[\leadsto \frac{z \cdot y}{\color{blue}{a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 36.0%

       \[\leadsto y \cdot \frac{z}{\color{blue}{a}} \]

   if -1.5999999999999999e106 < z < 1.12e83

   1. Initial program 64.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 57.2

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 57.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{t}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{t} + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\left(y - x\right) \cdot \left(z - t\right)}{t}\right)\right)} + x \]

      3. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - t}{t}}\right)\right) + x \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - t}{t}} + x \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - t}{t} + x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - t}{t}, x\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(y - x\right)\right)}, \frac{z - t}{t}, x\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-\left(y - x\right)}, \frac{z - t}{t}, x\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(-\color{blue}{\left(y - x\right)}, \frac{z - t}{t}, x\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-\left(y - x\right), \color{blue}{\frac{z - t}{t}}, x\right) \]

      11. lower--.f64 27.2

          \[\leadsto \mathsf{fma}\left(-\left(y - x\right), \frac{\color{blue}{z - t}}{t}, x\right) \]

   7. Applied rewrites 27.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\left(y - x\right), \frac{z - t}{t}, x\right)} \]

   8. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{-1 \cdot \left(x - y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 19.3%

       \[\leadsto x - \color{blue}{\left(x - y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto x - -1 \cdot y \]
   12. Step-by-step derivation

   13. Applied rewrites 42.3%

       \[\leadsto x - \left(-y\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+106} \lor \neg \left(z \leq 1.12 \cdot 10^{+83}\right):\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 39.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+106}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{+83}:\\ \;\;\;\;x - \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.6e+106)
   (* y (/ z a))
   (if (<= z 1.12e+83) (- x (- y)) (* (/ y a) z))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.6e+106) {
		tmp = y * (z / a);
	} else if (z <= 1.12e+83) {
		tmp = x - -y;
	} else {
		tmp = (y / a) * z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.6d+106)) then
        tmp = y * (z / a)
    else if (z <= 1.12d+83) then
        tmp = x - -y
    else
        tmp = (y / a) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.6e+106) {
		tmp = y * (z / a);
	} else if (z <= 1.12e+83) {
		tmp = x - -y;
	} else {
		tmp = (y / a) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.6e+106:
		tmp = y * (z / a)
	elif z <= 1.12e+83:
		tmp = x - -y
	else:
		tmp = (y / a) * z
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.6e+106)
		tmp = Float64(y * Float64(z / a));
	elseif (z <= 1.12e+83)
		tmp = Float64(x - Float64(-y));
	else
		tmp = Float64(Float64(y / a) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.6e+106)
		tmp = y * (z / a);
	elseif (z <= 1.12e+83)
		tmp = x - -y;
	else
		tmp = (y / a) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.6e+106], N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.12e+83], N[(x - (-y)), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.5999999999999999e106

   1. Initial program 69.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(y - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{a}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y - x}{a} \cdot z} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - x}{a}}, z, x\right) \]

      6. lower--.f64 51.7

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - x}}{a}, z, x\right) \]

   5. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{y \cdot z}{\color{blue}{a}} \]
   7. Step-by-step derivation

   8. Applied rewrites 26.5%

      \[\leadsto \frac{z \cdot y}{\color{blue}{a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 30.6%

       \[\leadsto y \cdot \frac{z}{\color{blue}{a}} \]

   if -1.5999999999999999e106 < z < 1.12e83

   1. Initial program 64.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 57.2

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 57.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{t}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{t} + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\left(y - x\right) \cdot \left(z - t\right)}{t}\right)\right)} + x \]

      3. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - t}{t}}\right)\right) + x \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - t}{t}} + x \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - t}{t} + x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - t}{t}, x\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(y - x\right)\right)}, \frac{z - t}{t}, x\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-\left(y - x\right)}, \frac{z - t}{t}, x\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(-\color{blue}{\left(y - x\right)}, \frac{z - t}{t}, x\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-\left(y - x\right), \color{blue}{\frac{z - t}{t}}, x\right) \]

      11. lower--.f64 27.2

          \[\leadsto \mathsf{fma}\left(-\left(y - x\right), \frac{\color{blue}{z - t}}{t}, x\right) \]

   7. Applied rewrites 27.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\left(y - x\right), \frac{z - t}{t}, x\right)} \]

   8. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{-1 \cdot \left(x - y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 19.3%

       \[\leadsto x - \color{blue}{\left(x - y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto x - -1 \cdot y \]
   12. Step-by-step derivation

   13. Applied rewrites 42.3%

       \[\leadsto x - \left(-y\right) \]

   if 1.12e83 < z

   1. Initial program 76.3%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(y - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{a}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y - x}{a} \cdot z} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - x}{a}}, z, x\right) \]

      6. lower--.f64 59.4

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - x}}{a}, z, x\right) \]

   5. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{y \cdot z}{\color{blue}{a}} \]
   7. Step-by-step derivation

   8. Applied rewrites 34.3%

      \[\leadsto \frac{z \cdot y}{\color{blue}{a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 42.7%

       \[\leadsto \frac{y}{a} \cdot z \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+106}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{+83}:\\ \;\;\;\;x - \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 37.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{+242} \lor \neg \left(t \leq 2.8 \cdot 10^{+125}\right):\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x - \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -2.2e+242) (not (<= t 2.8e+125))) y (- x (- y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -2.2e+242) || !(t <= 2.8e+125)) {
		tmp = y;
	} else {
		tmp = x - -y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((t <= (-2.2d+242)) .or. (.not. (t <= 2.8d+125))) then
        tmp = y
    else
        tmp = x - -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -2.2e+242) || !(t <= 2.8e+125)) {
		tmp = y;
	} else {
		tmp = x - -y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -2.2e+242) or not (t <= 2.8e+125):
		tmp = y
	else:
		tmp = x - -y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -2.2e+242) || !(t <= 2.8e+125))
		tmp = y;
	else
		tmp = Float64(x - Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -2.2e+242) || ~((t <= 2.8e+125)))
		tmp = y;
	else
		tmp = x - -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -2.2e+242], N[Not[LessEqual[t, 2.8e+125]], $MachinePrecision]], y, N[(x - (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.19999999999999999e242 or 2.8000000000000001e125 < t

   1. Initial program 29.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 14.2

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 14.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{t}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{t} + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\left(y - x\right) \cdot \left(z - t\right)}{t}\right)\right)} + x \]

      3. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - t}{t}}\right)\right) + x \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - t}{t}} + x \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - t}{t} + x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - t}{t}, x\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(y - x\right)\right)}, \frac{z - t}{t}, x\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-\left(y - x\right)}, \frac{z - t}{t}, x\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(-\color{blue}{\left(y - x\right)}, \frac{z - t}{t}, x\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-\left(y - x\right), \color{blue}{\frac{z - t}{t}}, x\right) \]

      11. lower--.f64 42.1

          \[\leadsto \mathsf{fma}\left(-\left(y - x\right), \frac{\color{blue}{z - t}}{t}, x\right) \]

   7. Applied rewrites 42.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\left(y - x\right), \frac{z - t}{t}, x\right)} \]

   8. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{-1 \cdot \left(x - y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 34.6%

       \[\leadsto x - \color{blue}{\left(x - y\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 56.6%

       \[\leadsto y + 0 \]

   if -2.19999999999999999e242 < t < 2.8000000000000001e125

   1. Initial program 77.9%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

      5. flip-- N/A

         \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

      14. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

      17. lower-+.f64 70.6

          \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

   4. Applied rewrites 70.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{t}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{t} + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\left(y - x\right) \cdot \left(z - t\right)}{t}\right)\right)} + x \]

      3. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - t}{t}}\right)\right) + x \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - t}{t}} + x \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - t}{t} + x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - t}{t}, x\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(y - x\right)\right)}, \frac{z - t}{t}, x\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-\left(y - x\right)}, \frac{z - t}{t}, x\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(-\color{blue}{\left(y - x\right)}, \frac{z - t}{t}, x\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-\left(y - x\right), \color{blue}{\frac{z - t}{t}}, x\right) \]

      11. lower--.f64 35.2

          \[\leadsto \mathsf{fma}\left(-\left(y - x\right), \frac{\color{blue}{z - t}}{t}, x\right) \]

   7. Applied rewrites 35.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\left(y - x\right), \frac{z - t}{t}, x\right)} \]

   8. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{-1 \cdot \left(x - y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 10.3%

       \[\leadsto x - \color{blue}{\left(x - y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto x - -1 \cdot y \]
   12. Step-by-step derivation

   13. Applied rewrites 30.7%

       \[\leadsto x - \left(-y\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{+242} \lor \neg \left(t \leq 2.8 \cdot 10^{+125}\right):\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x - \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 25.7% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ y \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 y)

C

double code(double x, double y, double z, double t, double a) {
	return y;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = y
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return y;
}

Python

def code(x, y, z, t, a):
	return y

Julia

function code(x, y, z, t, a)
	return y
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = y;
end

Wolfram

code[x_, y_, z_, t_, a_] := y

Derivation

1. Initial program 67.8%

   \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

   3. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

   4. lift--.f64 N/A

      \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{a - t}} + x \]

   5. flip-- N/A

      \[\leadsto \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{\frac{a \cdot a - t \cdot t}{a + t}}} + x \]

   6. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t} \cdot \left(a + t\right)} + x \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\left(a + t\right) \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}} + x \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right)} \]

   9. lower-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{a + t}, \frac{\left(y - x\right) \cdot \left(z - t\right)}{a \cdot a - t \cdot t}, x\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a + t, \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a \cdot a - t \cdot t}, x\right) \]

   11. associate-/l* N/A

       \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a + t, \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

   13. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \color{blue}{\frac{z - t}{a \cdot a - t \cdot t}}, x\right) \]

   14. difference-of-squares N/A

       \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

   15. lift--.f64 N/A

       \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \color{blue}{\left(a - t\right)}}, x\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right) \cdot \left(a - t\right)}}, x\right) \]

   17. lower-+.f64 58.9

       \[\leadsto \mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\color{blue}{\left(a + t\right)} \cdot \left(a - t\right)}, x\right) \]

4. Applied rewrites 58.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a + t, \left(y - x\right) \cdot \frac{z - t}{\left(a + t\right) \cdot \left(a - t\right)}, x\right)} \]

5. Taylor expanded in a around 0

   \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{t}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(y - x\right) \cdot \left(z - t\right)}{t} + x} \]

   2. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\left(y - x\right) \cdot \left(z - t\right)}{t}\right)\right)} + x \]

   3. associate-/l* N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - t}{t}}\right)\right) + x \]

   4. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - t}{t}} + x \]

   5. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - t}{t} + x \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - t}{t}, x\right)} \]

   7. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(y - x\right)\right)}, \frac{z - t}{t}, x\right) \]

   8. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{-\left(y - x\right)}, \frac{z - t}{t}, x\right) \]

   9. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(-\color{blue}{\left(y - x\right)}, \frac{z - t}{t}, x\right) \]

   10. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(-\left(y - x\right), \color{blue}{\frac{z - t}{t}}, x\right) \]

   11. lower--.f64 36.7

       \[\leadsto \mathsf{fma}\left(-\left(y - x\right), \frac{\color{blue}{z - t}}{t}, x\right) \]

7. Applied rewrites 36.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-\left(y - x\right), \frac{z - t}{t}, x\right)} \]

8. Taylor expanded in z around 0

   \[\leadsto x + \color{blue}{-1 \cdot \left(x - y\right)} \]
9. Step-by-step derivation

10. Applied rewrites 15.3%

    \[\leadsto x - \color{blue}{\left(x - y\right)} \]
11. Step-by-step derivation

12. Applied rewrites 21.2%

    \[\leadsto y + 0 \]

13. Final simplification 21.2%

    \[\leadsto y \]
14. Add Preprocessing

Developer Target 1: 86.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;a < -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a < 3.774403170083174 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))
   (if (< a -1.6153062845442575e-142)
     t_1
     (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	double tmp;
	if (a < -1.6153062845442575e-142) {
		tmp = t_1;
	} else if (a < 3.774403170083174e-182) {
		tmp = y - ((z / t) * (y - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (((y - x) / 1.0d0) * ((z - t) / (a - t)))
    if (a < (-1.6153062845442575d-142)) then
        tmp = t_1
    else if (a < 3.774403170083174d-182) then
        tmp = y - ((z / t) * (y - x))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	double tmp;
	if (a < -1.6153062845442575e-142) {
		tmp = t_1;
	} else if (a < 3.774403170083174e-182) {
		tmp = y - ((z / t) * (y - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)))
	tmp = 0
	if a < -1.6153062845442575e-142:
		tmp = t_1
	elif a < 3.774403170083174e-182:
		tmp = y - ((z / t) * (y - x))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) / 1.0) * Float64(Float64(z - t) / Float64(a - t))))
	tmp = 0.0
	if (a < -1.6153062845442575e-142)
		tmp = t_1;
	elseif (a < 3.774403170083174e-182)
		tmp = Float64(y - Float64(Float64(z / t) * Float64(y - x)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	tmp = 0.0;
	if (a < -1.6153062845442575e-142)
		tmp = t_1;
	elseif (a < 3.774403170083174e-182)
		tmp = y - ((z / t) * (y - x));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] / 1.0), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[a, -1.6153062845442575e-142], t$95$1, If[Less[a, 3.774403170083174e-182], N[(y - N[(N[(z / t), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024359 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< a -646122513817703/4000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))) (if (< a 1887201585041587/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))))))

  (+ x (/ (* (- y x) (- z t)) (- a t))))