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

Percentage Accurate: 68.1% → 89.7%

Time: 4.9s

Alternatives: 16

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: 68.1% 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: 89.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y - \left(y - x\right) \cdot \frac{z - a}{t}\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{t - a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- y (* (- y x) (/ (- z a) t)))))
   (if (<= t -4.5e+150)
     t_1
     (if (<= t 1.15e+139) (fma (/ (- t z) (- t a)) (- y x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y - ((y - x) * ((z - a) / t));
	double tmp;
	if (t <= -4.5e+150) {
		tmp = t_1;
	} else if (t <= 1.15e+139) {
		tmp = fma(((t - z) / (t - a)), (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y - Float64(Float64(y - x) * Float64(Float64(z - a) / t)))
	tmp = 0.0
	if (t <= -4.5e+150)
		tmp = t_1;
	elseif (t <= 1.15e+139)
		tmp = fma(Float64(Float64(t - z) / Float64(t - a)), Float64(y - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y - N[(N[(y - x), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.5e+150], t$95$1, If[LessEqual[t, 1.15e+139], N[(N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.5e150 or 1.15e139 < t

   1. Initial program 68.1%

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 84.0

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

   3. Applied rewrites 84.0%

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

   4. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 46.0

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

   6. Applied rewrites 46.0%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. mult-flip N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. mult-flip-rev N/A

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

      18. lower-/.f64 N/A

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

      19. lower--.f64 52.8

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

   8. Applied rewrites 52.8%

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

   if -4.5e150 < t < 1.15e139

   1. Initial program 68.1%

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 84.0

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

   3. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{t - a}, y - x, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - z}{t - a}, y, x\right)\\ \mathbf{if}\;a \leq -2.85 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4.3 \cdot 10^{+43}:\\ \;\;\;\;y - \left(y - x\right) \cdot \frac{z - a}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- t z) (- t a)) y x)))
   (if (<= a -2.85e+29)
     t_1
     (if (<= a 4.3e+43) (- y (* (- y x) (/ (- z a) t))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - z) / (t - a)), y, x);
	double tmp;
	if (a <= -2.85e+29) {
		tmp = t_1;
	} else if (a <= 4.3e+43) {
		tmp = y - ((y - x) * ((z - a) / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - z) / Float64(t - a)), y, x)
	tmp = 0.0
	if (a <= -2.85e+29)
		tmp = t_1;
	elseif (a <= 4.3e+43)
		tmp = Float64(y - Float64(Float64(y - 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[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[a, -2.85e+29], t$95$1, If[LessEqual[a, 4.3e+43], N[(y - N[(N[(y - x), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.85e29 or 4.3e43 < a

   1. Initial program 68.1%

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 84.0

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

   3. Applied rewrites 84.0%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 67.5%

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

   if -2.85e29 < a < 4.3e43

   1. Initial program 68.1%

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 84.0

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

   3. Applied rewrites 84.0%

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

   4. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 46.0

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

   6. Applied rewrites 46.0%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. mult-flip N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. mult-flip-rev N/A

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

      18. lower-/.f64 N/A

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

      19. lower--.f64 52.8

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

   8. Applied rewrites 52.8%

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

Alternative 3: 72.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - y}{t - a} \cdot z\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+198}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{t - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{z}{t - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.25e+55)
   (* (/ (- x y) (- t a)) z)
   (if (<= z 2.2e+198)
     (fma (/ (- t z) (- t a)) y x)
     (* (- x y) (/ z (- t a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.25e+55) {
		tmp = ((x - y) / (t - a)) * z;
	} else if (z <= 2.2e+198) {
		tmp = fma(((t - z) / (t - a)), y, x);
	} else {
		tmp = (x - y) * (z / (t - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.25e+55)
		tmp = Float64(Float64(Float64(x - y) / Float64(t - a)) * z);
	elseif (z <= 2.2e+198)
		tmp = fma(Float64(Float64(t - z) / Float64(t - a)), y, x);
	else
		tmp = Float64(Float64(x - y) * Float64(z / Float64(t - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.25e+55], N[(N[(N[(x - y), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 2.2e+198], N[(N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(z / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.24999999999999999e55

   1. Initial program 68.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 80.3

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

   3. Applied rewrites 80.3%

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

   4. Taylor expanded in z around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 38.1

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

   6. Applied rewrites 38.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift--.f64 N/A

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

      5. sub-div N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 42.4

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

      11. lift--.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. sub-div N/A

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

      15. lift--.f64 N/A

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

      16. lower-/.f64 42.6

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

   8. Applied rewrites 42.6%

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

   if -2.24999999999999999e55 < z < 2.2e198

   1. Initial program 68.1%

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 84.0

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

   3. Applied rewrites 84.0%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 67.5%

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

   if 2.2e198 < z

   1. Initial program 68.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 80.3

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

   3. Applied rewrites 80.3%

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

   4. Taylor expanded in z around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 38.1

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

   6. Applied rewrites 38.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 43.5

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

   8. Applied rewrites 43.5%

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

Alternative 4: 65.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+83}:\\ \;\;\;\;\frac{x - y}{t - a} \cdot z\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-78}:\\ \;\;\;\;\frac{z - t}{a - t} \cdot y\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{t - a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{z}{t - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.7e+83)
   (* (/ (- x y) (- t a)) z)
   (if (<= z -1.65e-78)
     (* (/ (- z t) (- a t)) y)
     (if (<= z 2.2e+140)
       (fma (/ t (- t a)) (- y x) x)
       (* (- x y) (/ z (- t a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.7e+83) {
		tmp = ((x - y) / (t - a)) * z;
	} else if (z <= -1.65e-78) {
		tmp = ((z - t) / (a - t)) * y;
	} else if (z <= 2.2e+140) {
		tmp = fma((t / (t - a)), (y - x), x);
	} else {
		tmp = (x - y) * (z / (t - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.7e+83)
		tmp = Float64(Float64(Float64(x - y) / Float64(t - a)) * z);
	elseif (z <= -1.65e-78)
		tmp = Float64(Float64(Float64(z - t) / Float64(a - t)) * y);
	elseif (z <= 2.2e+140)
		tmp = fma(Float64(t / Float64(t - a)), Float64(y - x), x);
	else
		tmp = Float64(Float64(x - y) * Float64(z / Float64(t - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.7e+83], N[(N[(N[(x - y), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, -1.65e-78], N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 2.2e+140], N[(N[(t / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(z / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.7000000000000002e83

   1. Initial program 68.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 80.3

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

   3. Applied rewrites 80.3%

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

   4. Taylor expanded in z around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 38.1

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

   6. Applied rewrites 38.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift--.f64 N/A

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

      5. sub-div N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 42.4

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

      11. lift--.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. sub-div N/A

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

      15. lift--.f64 N/A

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

      16. lower-/.f64 42.6

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

   8. Applied rewrites 42.6%

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

   if -3.7000000000000002e83 < z < -1.64999999999999991e-78

   1. Initial program 68.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 40.1

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

   4. Applied rewrites 40.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{a - t} \]

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \]

      9. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \]

      10. frac-2neg N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 51.7

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

      14. lift-/.f64 N/A

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

      15. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      16. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      17. sub-negate-rev N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      18. lift--.f64 N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      19. lift--.f64 N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      20. sub-negate-rev N/A

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

      21. lift--.f64 N/A

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

      22. lower-/.f64 51.7

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

   6. Applied rewrites 51.7%

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

   if -1.64999999999999991e-78 < z < 2.1999999999999998e140

   1. Initial program 68.1%

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 84.0

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

   3. Applied rewrites 84.0%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 46.7%

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

   if 2.1999999999999998e140 < z

   1. Initial program 68.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 80.3

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

   3. Applied rewrites 80.3%

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

   4. Taylor expanded in z around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 38.1

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

   6. Applied rewrites 38.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 43.5

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

   8. Applied rewrites 43.5%

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

Alternative 5: 64.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t} \cdot y\\ \mathbf{if}\;y \leq -0.065:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t - a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- z t) (- a t)) y)))
   (if (<= y -0.065)
     t_1
     (if (<= y 1.4e-29) (fma (/ x (- t a)) (- z t) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) / (a - t)) * y;
	double tmp;
	if (y <= -0.065) {
		tmp = t_1;
	} else if (y <= 1.4e-29) {
		tmp = fma((x / (t - a)), (z - t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - t) / Float64(a - t)) * y)
	tmp = 0.0
	if (y <= -0.065)
		tmp = t_1;
	elseif (y <= 1.4e-29)
		tmp = fma(Float64(x / Float64(t - a)), Float64(z - t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -0.065], t$95$1, If[LessEqual[y, 1.4e-29], N[(N[(x / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.065000000000000002 or 1.4000000000000001e-29 < y

   1. Initial program 68.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 40.1

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

   4. Applied rewrites 40.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{a - t} \]

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \]

      9. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \]

      10. frac-2neg N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 51.7

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

      14. lift-/.f64 N/A

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

      15. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      16. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      17. sub-negate-rev N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      18. lift--.f64 N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      19. lift--.f64 N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      20. sub-negate-rev N/A

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

      21. lift--.f64 N/A

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

      22. lower-/.f64 51.7

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

   6. Applied rewrites 51.7%

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

   if -0.065000000000000002 < y < 1.4000000000000001e-29

   1. Initial program 68.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 80.3

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

   3. Applied rewrites 80.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 41.4

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

   6. Applied rewrites 41.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{t - a}}, z - t, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 64.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{if}\;a \leq -2.7 \cdot 10^{+98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{+79}:\\ \;\;\;\;\frac{z - t}{a - t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z a) (- y x) x)))
   (if (<= a -2.7e+98) t_1 (if (<= a 8.5e+79) (* (/ (- z t) (- a 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 <= -2.7e+98) {
		tmp = t_1;
	} else if (a <= 8.5e+79) {
		tmp = ((z - t) / (a - 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 <= -2.7e+98)
		tmp = t_1;
	elseif (a <= 8.5e+79)
		tmp = Float64(Float64(Float64(z - t) / Float64(a - t)) * 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, -2.7e+98], t$95$1, If[LessEqual[a, 8.5e+79], N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.7e98 or 8.4999999999999998e79 < a

   1. Initial program 68.1%

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 84.0

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

   3. Applied rewrites 84.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 49.4

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

   6. Applied rewrites 49.4%

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

   if -2.7e98 < a < 8.4999999999999998e79

   1. Initial program 68.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 40.1

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

   4. Applied rewrites 40.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{a - t} \]

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \]

      9. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \]

      10. frac-2neg N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 51.7

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

      14. lift-/.f64 N/A

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

      15. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      16. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      17. sub-negate-rev N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      18. lift--.f64 N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      19. lift--.f64 N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      20. sub-negate-rev N/A

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

      21. lift--.f64 N/A

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

      22. lower-/.f64 51.7

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

   6. Applied rewrites 51.7%

      \[\leadsto \frac{z - t}{a - t} \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 61.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{if}\;a \leq -5 \cdot 10^{+97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{+79}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z a) (- y x) x)))
   (if (<= a -5e+97) t_1 (if (<= a 8.5e+79) (* (- z t) (/ y (- a t))) 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 <= -5e+97) {
		tmp = t_1;
	} else if (a <= 8.5e+79) {
		tmp = (z - t) * (y / (a - t));
	} 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 <= -5e+97)
		tmp = t_1;
	elseif (a <= 8.5e+79)
		tmp = Float64(Float64(z - t) * Float64(y / Float64(a - t)));
	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, -5e+97], t$95$1, If[LessEqual[a, 8.5e+79], N[(N[(z - t), $MachinePrecision] * N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.99999999999999999e97 or 8.4999999999999998e79 < a

   1. Initial program 68.1%

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 84.0

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

   3. Applied rewrites 84.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 49.4

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

   6. Applied rewrites 49.4%

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

   if -4.99999999999999999e97 < a < 8.4999999999999998e79

   1. Initial program 68.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 40.1

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

   4. Applied rewrites 40.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 46.5

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

   6. Applied rewrites 46.5%

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

Alternative 8: 60.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - z}{t}, y, x\right)\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{-57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- t z) t) y x)))
   (if (<= t -9.5e-57) t_1 (if (<= t 4e+14) (fma (/ z a) (- y x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - z) / t), y, x);
	double tmp;
	if (t <= -9.5e-57) {
		tmp = t_1;
	} else if (t <= 4e+14) {
		tmp = fma((z / a), (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - z) / t), y, x)
	tmp = 0.0
	if (t <= -9.5e-57)
		tmp = t_1;
	elseif (t <= 4e+14)
		tmp = fma(Float64(z / a), Float64(y - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t - z), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -9.5e-57], t$95$1, If[LessEqual[t, 4e+14], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -9.5000000000000005e-57 or 4e14 < t

   1. Initial program 68.1%

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 84.0

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

   3. Applied rewrites 84.0%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 38.2

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

   6. Applied rewrites 38.2%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 41.9%

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

   if -9.5000000000000005e-57 < t < 4e14

   1. Initial program 68.1%

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 84.0

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

   3. Applied rewrites 84.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 49.4

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

   6. Applied rewrites 49.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{a}}, y - x, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 57.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{if}\;t \leq -1.12 \cdot 10^{+149}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma 1.0 (- y x) x)))
   (if (<= t -1.12e+149)
     t_1
     (if (<= t 9.8e+120) (fma (/ z a) (- y x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(1.0, (y - x), x);
	double tmp;
	if (t <= -1.12e+149) {
		tmp = t_1;
	} else if (t <= 9.8e+120) {
		tmp = fma((z / a), (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(1.0, Float64(y - x), x)
	tmp = 0.0
	if (t <= -1.12e+149)
		tmp = t_1;
	elseif (t <= 9.8e+120)
		tmp = fma(Float64(z / a), Float64(y - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -1.12e+149], t$95$1, If[LessEqual[t, 9.8e+120], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.11999999999999992e149 or 9.80000000000000021e120 < t

   1. Initial program 68.1%

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 84.0

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

   3. Applied rewrites 84.0%

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

   4. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, y - x, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 19.4%

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

   if -1.11999999999999992e149 < t < 9.80000000000000021e120

   1. Initial program 68.1%

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 84.0

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

   3. Applied rewrites 84.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 49.4

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

   6. Applied rewrites 49.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{a}}, y - x, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 32.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+54}:\\ \;\;\;\;z \cdot \frac{x - y}{t}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(x - y\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.6e+54)
   (* z (/ (- x y) t))
   (if (<= z 1.6e+158) (fma 1.0 (- y x) x) (/ (* z (- x y)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.6e+54) {
		tmp = z * ((x - y) / t);
	} else if (z <= 1.6e+158) {
		tmp = fma(1.0, (y - x), x);
	} else {
		tmp = (z * (x - y)) / t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.6e+54)
		tmp = Float64(z * Float64(Float64(x - y) / t));
	elseif (z <= 1.6e+158)
		tmp = fma(1.0, Float64(y - x), x);
	else
		tmp = Float64(Float64(z * Float64(x - y)) / t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.6e+54], N[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.6e+158], N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(z * N[(x - y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.59999999999999988e54

   1. Initial program 68.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 80.3

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

   3. Applied rewrites 80.3%

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

   4. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 42.4

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

   6. Applied rewrites 42.4%

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

   7. Taylor expanded in t around inf

      \[\leadsto z \cdot \frac{x - y}{\color{blue}{t}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower--.f64 25.5

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

   9. Applied rewrites 25.5%

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

   if -4.59999999999999988e54 < z < 1.59999999999999997e158

   1. Initial program 68.1%

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 84.0

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

   3. Applied rewrites 84.0%

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

   4. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, y - x, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 19.4%

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

   if 1.59999999999999997e158 < z

   1. Initial program 68.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 80.3

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

   3. Applied rewrites 80.3%

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

   4. Taylor expanded in z around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 38.1

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

   6. Applied rewrites 38.1%

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

   7. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 23.7

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

   9. Applied rewrites 23.7%

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

Alternative 11: 31.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \left(x - y\right)}{t}\\ \mathbf{if}\;z \leq -4.6 \cdot 10^{+54}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* z (- x y)) t)))
   (if (<= z -4.6e+54) t_1 (if (<= z 1.6e+158) (fma 1.0 (- y x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * (x - y)) / t;
	double tmp;
	if (z <= -4.6e+54) {
		tmp = t_1;
	} else if (z <= 1.6e+158) {
		tmp = fma(1.0, (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * Float64(x - y)) / t)
	tmp = 0.0
	if (z <= -4.6e+54)
		tmp = t_1;
	elseif (z <= 1.6e+158)
		tmp = fma(1.0, Float64(y - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * N[(x - y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[z, -4.6e+54], t$95$1, If[LessEqual[z, 1.6e+158], N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.59999999999999988e54 or 1.59999999999999997e158 < z

   1. Initial program 68.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. 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. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 80.3

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

   3. Applied rewrites 80.3%

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

   4. Taylor expanded in z around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 38.1

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

   6. Applied rewrites 38.1%

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

   7. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 23.7

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

   9. Applied rewrites 23.7%

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

   if -4.59999999999999988e54 < z < 1.59999999999999997e158

   1. Initial program 68.1%

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 84.0

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

   3. Applied rewrites 84.0%

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

   4. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, y - x, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 19.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, y - x, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 30.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.3 \cdot 10^{-56}:\\ \;\;\;\;\frac{y \cdot z}{a - t}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-94}:\\ \;\;\;\;\frac{x \cdot \left(z - a\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -7.3e-56)
   (/ (* y z) (- a t))
   (if (<= y 9.5e-94) (/ (* x (- z a)) t) (fma 1.0 (- y x) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -7.3e-56) {
		tmp = (y * z) / (a - t);
	} else if (y <= 9.5e-94) {
		tmp = (x * (z - a)) / t;
	} else {
		tmp = fma(1.0, (y - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -7.3e-56)
		tmp = Float64(Float64(y * z) / Float64(a - t));
	elseif (y <= 9.5e-94)
		tmp = Float64(Float64(x * Float64(z - a)) / t);
	else
		tmp = fma(1.0, Float64(y - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -7.3e-56], N[(N[(y * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.5e-94], N[(N[(x * N[(z - a), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.30000000000000045e-56

   1. Initial program 68.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 40.1

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

   4. Applied rewrites 40.1%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{y \cdot z}{\color{blue}{a} - t} \]
   6. Step-by-step derivation

      1. lower-*.f64 21.6

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

   7. Applied rewrites 21.6%

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

   if -7.30000000000000045e-56 < y < 9.4999999999999997e-94

   1. Initial program 68.1%

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 84.0

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

   3. Applied rewrites 84.0%

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

   4. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 46.0

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

   6. Applied rewrites 46.0%

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

   7. Taylor expanded in x around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 19.8

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

   9. Applied rewrites 19.8%

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

   if 9.4999999999999997e-94 < y

   1. Initial program 68.1%

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 84.0

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

   3. Applied rewrites 84.0%

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

   4. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, y - x, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 19.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, y - x, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 29.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+23}:\\ \;\;\;\;\frac{y \cdot z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma 1.0 (- y x) x)))
   (if (<= t -3.8e+80) t_1 (if (<= t 1.1e+23) (/ (* y z) (- a t)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(1.0, (y - x), x);
	double tmp;
	if (t <= -3.8e+80) {
		tmp = t_1;
	} else if (t <= 1.1e+23) {
		tmp = (y * z) / (a - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(1.0, Float64(y - x), x)
	tmp = 0.0
	if (t <= -3.8e+80)
		tmp = t_1;
	elseif (t <= 1.1e+23)
		tmp = Float64(Float64(y * z) / Float64(a - t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -3.8e+80], t$95$1, If[LessEqual[t, 1.1e+23], N[(N[(y * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.79999999999999997e80 or 1.10000000000000004e23 < t

   1. Initial program 68.1%

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 84.0

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

   3. Applied rewrites 84.0%

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

   4. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, y - x, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 19.4%

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

   if -3.79999999999999997e80 < t < 1.10000000000000004e23

   1. Initial program 68.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 40.1

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

   4. Applied rewrites 40.1%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{y \cdot z}{\color{blue}{a} - t} \]
   6. Step-by-step derivation

      1. lower-*.f64 21.6

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

   7. Applied rewrites 21.6%

      \[\leadsto \frac{y \cdot z}{\color{blue}{a} - t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 28.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 26.5:\\ \;\;\;\;\frac{z}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma 1.0 (- y x) x)))
   (if (<= t -1.6e+19) t_1 (if (<= t 26.5) (* (/ z a) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(1.0, (y - x), x);
	double tmp;
	if (t <= -1.6e+19) {
		tmp = t_1;
	} else if (t <= 26.5) {
		tmp = (z / a) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(1.0, Float64(y - x), x)
	tmp = 0.0
	if (t <= -1.6e+19)
		tmp = t_1;
	elseif (t <= 26.5)
		tmp = Float64(Float64(z / a) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -1.6e+19], t$95$1, If[LessEqual[t, 26.5], N[(N[(z / a), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.6e19 or 26.5 < t

   1. Initial program 68.1%

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 84.0

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

   3. Applied rewrites 84.0%

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

   4. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, y - x, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 19.4%

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

   if -1.6e19 < t < 26.5

   1. Initial program 68.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 40.1

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

   4. Applied rewrites 40.1%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{y \cdot z}{\color{blue}{a}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{a} \]

      2. lower-*.f64 17.1

         \[\leadsto \frac{y \cdot z}{a} \]

   7. Applied rewrites 17.1%

      \[\leadsto \frac{y \cdot z}{\color{blue}{a}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{a} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{a} \]

      3. associate-/l* N/A

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

      4. *-commutative N/A

         \[\leadsto \frac{z}{a} \cdot y \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{z}{a} \cdot y \]

      6. lower-/.f64 19.5

         \[\leadsto \frac{z}{a} \cdot y \]

   9. Applied rewrites 19.5%

      \[\leadsto \frac{z}{a} \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 19.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{z}{a} \cdot y \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return (z / a) * 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 = (z / a) * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

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

2. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower--.f64 40.1

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

4. Applied rewrites 40.1%

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

5. Taylor expanded in t around 0

   \[\leadsto \frac{y \cdot z}{\color{blue}{a}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{y \cdot z}{a} \]

   2. lower-*.f64 17.1

      \[\leadsto \frac{y \cdot z}{a} \]

7. Applied rewrites 17.1%

   \[\leadsto \frac{y \cdot z}{\color{blue}{a}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{y \cdot z}{a} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{y \cdot z}{a} \]

   3. associate-/l* N/A

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

   4. *-commutative N/A

      \[\leadsto \frac{z}{a} \cdot y \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{z}{a} \cdot y \]

   6. lower-/.f64 19.5

      \[\leadsto \frac{z}{a} \cdot y \]

9. Applied rewrites 19.5%

   \[\leadsto \frac{z}{a} \cdot y \]
10. Add Preprocessing

Alternative 16: 18.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ z \cdot \frac{y}{a} \end{array} \]

FPCore

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

C

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

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 = z * (y / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

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

2. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower--.f64 40.1

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

4. Applied rewrites 40.1%

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

5. Taylor expanded in t around 0

   \[\leadsto \frac{y \cdot z}{\color{blue}{a}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{y \cdot z}{a} \]

   2. lower-*.f64 17.1

      \[\leadsto \frac{y \cdot z}{a} \]

7. Applied rewrites 17.1%

   \[\leadsto \frac{y \cdot z}{\color{blue}{a}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{y \cdot z}{a} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{y \cdot z}{a} \]

   3. *-commutative N/A

      \[\leadsto \frac{z \cdot y}{a} \]

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 18.6

      \[\leadsto z \cdot \frac{y}{a} \]

9. Applied rewrites 18.6%

   \[\leadsto z \cdot \frac{y}{\color{blue}{a}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025155 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64
  (+ x (/ (* (- y x) (- z t)) (- a t))))