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

Percentage Accurate: 68.1% → 90.8%

Time: 5.3s

Alternatives: 16

Speedup: 0.9×

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: 90.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - y}{t}, z, y\right) - \left(-a \cdot \frac{y - x}{t}\right)\\ \mathbf{if}\;t \leq -7 \cdot 10^{+113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+207}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (fma (/ (- x y) t) z y) (- (* a (/ (- y x) t))))))
   (if (<= t -7e+113)
     t_1
     (if (<= t 5e+207) (fma (- y x) (/ (- z t) (- a t)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - y) / t), z, y) - -(a * ((y - x) / t));
	double tmp;
	if (t <= -7e+113) {
		tmp = t_1;
	} else if (t <= 5e+207) {
		tmp = fma((y - x), ((z - t) / (a - t)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(fma(Float64(Float64(x - y) / t), z, y) - Float64(-Float64(a * Float64(Float64(y - x) / t))))
	tmp = 0.0
	if (t <= -7e+113)
		tmp = t_1;
	elseif (t <= 5e+207)
		tmp = fma(Float64(y - x), Float64(Float64(z - t) / Float64(a - t)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * z + y), $MachinePrecision] - (-N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[t, -7e+113], t$95$1, If[LessEqual[t, 5e+207], N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.0000000000000001e113 or 4.9999999999999999e207 < t

   1. Initial program 68.1%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift--.f64 48.8

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

   7. Applied rewrites 48.8%

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

   if -7.0000000000000001e113 < t < 4.9999999999999999e207

   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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 84.2

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

   3. Applied rewrites 84.2%

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

Alternative 2: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-290}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\left(-\frac{\left(y - x\right) \cdot \left(z - a\right)}{t}\right) + y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y x) (/ (- z t) (- a t)) x))
        (t_2 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_2 -1e-290)
     t_1
     (if (<= t_2 0.0) (+ (- (/ (* (- y x) (- z a)) t)) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - x), ((z - t) / (a - t)), x);
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -1e-290) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = -(((y - x) * (z - a)) / t) + y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - x), Float64(Float64(z - t) / Float64(a - t)), x)
	t_2 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= -1e-290)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(-Float64(Float64(Float64(y - x) * Float64(z - a)) / t)) + y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-290], t$95$1, If[LessEqual[t$95$2, 0.0], N[((-N[(N[(N[(y - x), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]) + y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -1.0000000000000001e-290 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 84.2

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

   3. Applied rewrites 84.2%

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

   if -1.0000000000000001e-290 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 68.1%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

Alternative 3: 76.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y - \frac{y - x}{t} \cdot z\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{+144}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a - t}, x\right)\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+62}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- y (* (/ (- y x) t) z))))
   (if (<= t -2.8e+144)
     t_1
     (if (<= t 4.5e-115)
       (fma (- y x) (/ z (- a t)) x)
       (if (<= t 3.8e+62) (+ x (/ (* y (- z t)) (- a t))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y - (((y - x) / t) * z);
	double tmp;
	if (t <= -2.8e+144) {
		tmp = t_1;
	} else if (t <= 4.5e-115) {
		tmp = fma((y - x), (z / (a - t)), x);
	} else if (t <= 3.8e+62) {
		tmp = x + ((y * (z - t)) / (a - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y - Float64(Float64(Float64(y - x) / t) * z))
	tmp = 0.0
	if (t <= -2.8e+144)
		tmp = t_1;
	elseif (t <= 4.5e-115)
		tmp = fma(Float64(y - x), Float64(z / Float64(a - t)), x);
	elseif (t <= 3.8e+62)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / Float64(a - t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y - N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.8e+144], t$95$1, If[LessEqual[t, 4.5e-115], N[(N[(y - x), $MachinePrecision] * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 3.8e+62], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.80000000000000007e144 or 3.79999999999999984e62 < t

   1. Initial program 68.1%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 47.5

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

   7. Applied rewrites 47.5%

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

   if -2.80000000000000007e144 < t < 4.50000000000000023e-115

   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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 84.2

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

   3. Applied rewrites 84.2%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 61.7

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

   6. Applied rewrites 61.7%

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

   if 4.50000000000000023e-115 < t < 3.79999999999999984e62

   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

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

   4. Applied rewrites 56.4%

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

Alternative 4: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y - \frac{y - x}{t} \cdot z\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{+144}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- y (* (/ (- y x) t) z))))
   (if (<= t -2.8e+144)
     t_1
     (if (<= t 4.0) (fma (- y x) (/ z (- a t)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y - (((y - x) / t) * z);
	double tmp;
	if (t <= -2.8e+144) {
		tmp = t_1;
	} else if (t <= 4.0) {
		tmp = fma((y - x), (z / (a - t)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y - Float64(Float64(Float64(y - x) / t) * z))
	tmp = 0.0
	if (t <= -2.8e+144)
		tmp = t_1;
	elseif (t <= 4.0)
		tmp = fma(Float64(y - x), Float64(z / Float64(a - t)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y - N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.8e+144], t$95$1, If[LessEqual[t, 4.0], N[(N[(y - x), $MachinePrecision] * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.80000000000000007e144 or 4 < t

   1. Initial program 68.1%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 47.5

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

   7. Applied rewrites 47.5%

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

   if -2.80000000000000007e144 < t < 4

   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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 84.2

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

   3. Applied rewrites 84.2%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 61.7

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

   6. Applied rewrites 61.7%

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

Alternative 5: 73.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)\\ \mathbf{if}\;a \leq -6.5 \cdot 10^{-46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{-19}:\\ \;\;\;\;y - \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y x) (/ (- z t) a) x)))
   (if (<= a -6.5e-46) t_1 (if (<= a 3.8e-19) (- y (* (/ (- y x) t) z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - x), ((z - t) / a), x);
	double tmp;
	if (a <= -6.5e-46) {
		tmp = t_1;
	} else if (a <= 3.8e-19) {
		tmp = y - (((y - x) / t) * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - x), Float64(Float64(z - t) / a), x)
	tmp = 0.0
	if (a <= -6.5e-46)
		tmp = t_1;
	elseif (a <= 3.8e-19)
		tmp = Float64(y - Float64(Float64(Float64(y - x) / t) * z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -6.5e-46], t$95$1, If[LessEqual[a, 3.8e-19], N[(y - N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.49999999999999966e-46 or 3.8e-19 < a

   1. Initial program 68.1%

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

   2. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 54.1

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

   4. Applied rewrites 54.1%

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

   if -6.49999999999999966e-46 < a < 3.8e-19

   1. Initial program 68.1%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 47.5

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

   7. Applied rewrites 47.5%

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

Alternative 6: 69.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{-17}:\\ \;\;\;\;\frac{z - t}{a} \cdot y + x\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{+41}:\\ \;\;\;\;y - \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.05e-17)
   (+ (* (/ (- z t) a) y) x)
   (if (<= a 1.25e+41) (- y (* (/ (- y x) t) z)) (fma z (/ (- y x) a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.05e-17) {
		tmp = (((z - t) / a) * y) + x;
	} else if (a <= 1.25e+41) {
		tmp = y - (((y - x) / t) * z);
	} else {
		tmp = fma(z, ((y - x) / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.05e-17)
		tmp = Float64(Float64(Float64(Float64(z - t) / a) * y) + x);
	elseif (a <= 1.25e+41)
		tmp = Float64(y - Float64(Float64(Float64(y - x) / t) * z));
	else
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.05e-17], N[(N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 1.25e+41], N[(y - N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.05e-17

   1. Initial program 68.1%

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

   2. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 54.1

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 46.2%

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

      1. lift-fma.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 46.2

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

   9. Applied rewrites 46.2%

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

   if -2.05e-17 < a < 1.25000000000000006e41

   1. Initial program 68.1%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 47.5

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

   7. Applied rewrites 47.5%

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

   if 1.25000000000000006e41 < a

   1. Initial program 68.1%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 48.6

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

   4. Applied rewrites 48.6%

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

Alternative 7: 69.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{+41}:\\ \;\;\;\;y - \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.05e-17)
   (fma y (/ (- z t) a) x)
   (if (<= a 1.25e+41) (- y (* (/ (- y x) t) z)) (fma z (/ (- y x) a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.05e-17) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (a <= 1.25e+41) {
		tmp = y - (((y - x) / t) * z);
	} else {
		tmp = fma(z, ((y - x) / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.05e-17)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	elseif (a <= 1.25e+41)
		tmp = Float64(y - Float64(Float64(Float64(y - x) / t) * z));
	else
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.05e-17], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 1.25e+41], N[(y - N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.05e-17

   1. Initial program 68.1%

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

   2. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 54.1

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 46.2%

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

   if -2.05e-17 < a < 1.25000000000000006e41

   1. Initial program 68.1%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 47.5

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

   7. Applied rewrites 47.5%

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

   if 1.25000000000000006e41 < a

   1. Initial program 68.1%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 48.6

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

   4. Applied rewrites 48.6%

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

Alternative 8: 57.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-164}:\\ \;\;\;\;\left(z - a\right) \cdot \frac{x}{t}\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-41}:\\ \;\;\;\;y - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -3.3e-46)
   (fma y (/ (- z t) a) x)
   (if (<= a -6e-164)
     (* (- z a) (/ x t))
     (if (<= a 9e-41) (- y (/ (* z y) t)) (fma z (/ (- y x) a) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -3.3e-46) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (a <= -6e-164) {
		tmp = (z - a) * (x / t);
	} else if (a <= 9e-41) {
		tmp = y - ((z * y) / t);
	} else {
		tmp = fma(z, ((y - x) / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -3.3e-46)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	elseif (a <= -6e-164)
		tmp = Float64(Float64(z - a) * Float64(x / t));
	elseif (a <= 9e-41)
		tmp = Float64(y - Float64(Float64(z * y) / t));
	else
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -3.3e-46], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, -6e-164], N[(N[(z - a), $MachinePrecision] * N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9e-41], N[(y - N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -3.30000000000000013e-46

   1. Initial program 68.1%

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

   2. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 54.1

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 46.2%

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

   if -3.30000000000000013e-46 < a < -6.0000000000000002e-164

   1. Initial program 68.1%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 19.5

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

   7. Applied rewrites 19.5%

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

      1. flip-- 19.5

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 21.3

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

   9. Applied rewrites 21.3%

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

   if -6.0000000000000002e-164 < a < 9e-41

   1. Initial program 68.1%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 32.5

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

   7. Applied rewrites 32.5%

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

   8. Taylor expanded in z around inf

      \[\leadsto y - \frac{y \cdot z}{t} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 33.7

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

   10. Applied rewrites 33.7%

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

   if 9e-41 < a

   1. Initial program 68.1%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 48.6

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

   4. Applied rewrites 48.6%

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

Alternative 9: 57.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{if}\;a \leq -3.5 \cdot 10^{-46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-164}:\\ \;\;\;\;\left(z - a\right) \cdot \frac{x}{t}\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-41}:\\ \;\;\;\;y - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ (- y x) a) x)))
   (if (<= a -3.5e-46)
     t_1
     (if (<= a -6e-164)
       (* (- z a) (/ x t))
       (if (<= a 9e-41) (- y (/ (* z y) t)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, ((y - x) / a), x);
	double tmp;
	if (a <= -3.5e-46) {
		tmp = t_1;
	} else if (a <= -6e-164) {
		tmp = (z - a) * (x / t);
	} else if (a <= 9e-41) {
		tmp = y - ((z * y) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(Float64(y - x) / a), x)
	tmp = 0.0
	if (a <= -3.5e-46)
		tmp = t_1;
	elseif (a <= -6e-164)
		tmp = Float64(Float64(z - a) * Float64(x / t));
	elseif (a <= 9e-41)
		tmp = Float64(y - Float64(Float64(z * y) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -3.5e-46], t$95$1, If[LessEqual[a, -6e-164], N[(N[(z - a), $MachinePrecision] * N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9e-41], N[(y - N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.5000000000000002e-46 or 9e-41 < a

   1. Initial program 68.1%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 48.6

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

   4. Applied rewrites 48.6%

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

   if -3.5000000000000002e-46 < a < -6.0000000000000002e-164

   1. Initial program 68.1%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 19.5

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

   7. Applied rewrites 19.5%

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

      1. flip-- 19.5

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 21.3

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

   9. Applied rewrites 21.3%

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

   if -6.0000000000000002e-164 < a < 9e-41

   1. Initial program 68.1%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 32.5

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

   7. Applied rewrites 32.5%

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

   8. Taylor expanded in z around inf

      \[\leadsto y - \frac{y \cdot z}{t} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 33.7

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

   10. Applied rewrites 33.7%

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

Alternative 10: 52.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{if}\;a \leq -3.5 \cdot 10^{-46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-164}:\\ \;\;\;\;\left(z - a\right) \cdot \frac{x}{t}\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{+40}:\\ \;\;\;\;y - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ y a) x)))
   (if (<= a -3.5e-46)
     t_1
     (if (<= a -6e-164)
       (* (- z a) (/ x t))
       (if (<= a 7.5e+40) (- y (/ (* z y) t)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, (y / a), x);
	double tmp;
	if (a <= -3.5e-46) {
		tmp = t_1;
	} else if (a <= -6e-164) {
		tmp = (z - a) * (x / t);
	} else if (a <= 7.5e+40) {
		tmp = y - ((z * y) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(y / a), x)
	tmp = 0.0
	if (a <= -3.5e-46)
		tmp = t_1;
	elseif (a <= -6e-164)
		tmp = Float64(Float64(z - a) * Float64(x / t));
	elseif (a <= 7.5e+40)
		tmp = Float64(y - Float64(Float64(z * y) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -3.5e-46], t$95$1, If[LessEqual[a, -6e-164], N[(N[(z - a), $MachinePrecision] * N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 7.5e+40], N[(y - N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.5000000000000002e-46 or 7.4999999999999996e40 < a

   1. Initial program 68.1%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 48.6

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

   4. Applied rewrites 48.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 40.6%

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

   if -3.5000000000000002e-46 < a < -6.0000000000000002e-164

   1. Initial program 68.1%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 19.5

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

   7. Applied rewrites 19.5%

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

      1. flip-- 19.5

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 21.3

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

   9. Applied rewrites 21.3%

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

   if -6.0000000000000002e-164 < a < 7.4999999999999996e40

   1. Initial program 68.1%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 32.5

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

   7. Applied rewrites 32.5%

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

   8. Taylor expanded in z around inf

      \[\leadsto y - \frac{y \cdot z}{t} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 33.7

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

   10. Applied rewrites 33.7%

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

Alternative 11: 52.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{if}\;a \leq -3.5 \cdot 10^{-46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-164}:\\ \;\;\;\;x \cdot \frac{z - a}{t}\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{+40}:\\ \;\;\;\;y - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ y a) x)))
   (if (<= a -3.5e-46)
     t_1
     (if (<= a -6e-164)
       (* x (/ (- z a) t))
       (if (<= a 7.5e+40) (- y (/ (* z y) t)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, (y / a), x);
	double tmp;
	if (a <= -3.5e-46) {
		tmp = t_1;
	} else if (a <= -6e-164) {
		tmp = x * ((z - a) / t);
	} else if (a <= 7.5e+40) {
		tmp = y - ((z * y) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(y / a), x)
	tmp = 0.0
	if (a <= -3.5e-46)
		tmp = t_1;
	elseif (a <= -6e-164)
		tmp = Float64(x * Float64(Float64(z - a) / t));
	elseif (a <= 7.5e+40)
		tmp = Float64(y - Float64(Float64(z * y) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -3.5e-46], t$95$1, If[LessEqual[a, -6e-164], N[(x * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 7.5e+40], N[(y - N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.5000000000000002e-46 or 7.4999999999999996e40 < a

   1. Initial program 68.1%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 48.6

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

   4. Applied rewrites 48.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 40.6%

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

   if -3.5000000000000002e-46 < a < -6.0000000000000002e-164

   1. Initial program 68.1%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 19.5

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

   7. Applied rewrites 19.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sub-div N/A

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

      7. lower-*.f64 N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 22.6

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

   9. Applied rewrites 22.6%

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

   if -6.0000000000000002e-164 < a < 7.4999999999999996e40

   1. Initial program 68.1%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 32.5

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

   7. Applied rewrites 32.5%

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

   8. Taylor expanded in z around inf

      \[\leadsto y - \frac{y \cdot z}{t} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 33.7

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

   10. Applied rewrites 33.7%

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

Alternative 12: 49.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{if}\;a \leq -4 \cdot 10^{-46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 8 \cdot 10^{+44}:\\ \;\;\;\;\frac{x - y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ y a) x)))
   (if (<= a -4e-46) t_1 (if (<= a 8e+44) (* (/ (- x y) t) z) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, (y / a), x);
	double tmp;
	if (a <= -4e-46) {
		tmp = t_1;
	} else if (a <= 8e+44) {
		tmp = ((x - y) / t) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(y / a), x)
	tmp = 0.0
	if (a <= -4e-46)
		tmp = t_1;
	elseif (a <= 8e+44)
		tmp = Float64(Float64(Float64(x - y) / t) * z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -4e-46], t$95$1, If[LessEqual[a, 8e+44], N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.00000000000000009e-46 or 8.0000000000000007e44 < a

   1. Initial program 68.1%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 48.6

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

   4. Applied rewrites 48.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 40.6%

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

   if -4.00000000000000009e-46 < a < 8.0000000000000007e44

   1. Initial program 68.1%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 25.3

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

   7. Applied rewrites 25.3%

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

Alternative 13: 48.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - x, 1, x\right)\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{+176}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+113}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y x) 1.0 x)))
   (if (<= t -1.6e+176) t_1 (if (<= t 2.7e+113) (fma z (/ y a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - x), 1.0, x);
	double tmp;
	if (t <= -1.6e+176) {
		tmp = t_1;
	} else if (t <= 2.7e+113) {
		tmp = fma(z, (y / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - x), 1.0, x)
	tmp = 0.0
	if (t <= -1.6e+176)
		tmp = t_1;
	elseif (t <= 2.7e+113)
		tmp = fma(z, Float64(y / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] * 1.0 + x), $MachinePrecision]}, If[LessEqual[t, -1.6e+176], t$95$1, If[LessEqual[t, 2.7e+113], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.5999999999999999e176 or 2.70000000000000011e113 < 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 84.2

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

   3. Applied rewrites 84.2%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 19.3%

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

   if -1.5999999999999999e176 < t < 2.70000000000000011e113

   1. Initial program 68.1%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 48.6

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

   4. Applied rewrites 48.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 40.6%

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

Alternative 14: 30.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - x, 1, x\right)\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{+166}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -7 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{elif}\;t \leq 0.94:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y x) 1.0 x)))
   (if (<= t -2.8e+166)
     t_1
     (if (<= t -7e-15) (* x (/ z t)) (if (<= t 0.94) (* y (/ z a)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - x), 1.0, x);
	double tmp;
	if (t <= -2.8e+166) {
		tmp = t_1;
	} else if (t <= -7e-15) {
		tmp = x * (z / t);
	} else if (t <= 0.94) {
		tmp = y * (z / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - x), 1.0, x)
	tmp = 0.0
	if (t <= -2.8e+166)
		tmp = t_1;
	elseif (t <= -7e-15)
		tmp = Float64(x * Float64(z / t));
	elseif (t <= 0.94)
		tmp = Float64(y * Float64(z / a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] * 1.0 + x), $MachinePrecision]}, If[LessEqual[t, -2.8e+166], t$95$1, If[LessEqual[t, -7e-15], N[(x * N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.94], N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.79999999999999996e166 or 0.93999999999999995 < 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 84.2

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

   3. Applied rewrites 84.2%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 19.3%

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

   if -2.79999999999999996e166 < t < -7.0000000000000001e-15

   1. Initial program 68.1%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 19.5

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

   7. Applied rewrites 19.5%

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

   8. Taylor expanded in z around inf

      \[\leadsto \frac{x \cdot z}{t} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 18.6

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

   10. Applied rewrites 18.6%

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

   if -7.0000000000000001e-15 < t < 0.93999999999999995

   1. Initial program 68.1%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 48.6

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

   4. Applied rewrites 48.6%

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

   5. Taylor expanded in x 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. *-commutative N/A

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

      3. lower-*.f64 16.6

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

   7. Applied rewrites 16.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 19.2

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

   9. Applied rewrites 19.2%

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

Alternative 15: 25.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z}{a}\\ \mathbf{if}\;y \leq -1.9 \cdot 10^{-92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-20}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ z a))))
   (if (<= y -1.9e-92) t_1 (if (<= y 5.2e-20) (* x (/ z t)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z / a);
	double tmp;
	if (y <= -1.9e-92) {
		tmp = t_1;
	} else if (y <= 5.2e-20) {
		tmp = x * (z / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y * (z / a)
    if (y <= (-1.9d-92)) then
        tmp = t_1
    else if (y <= 5.2d-20) then
        tmp = x * (z / t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z / a);
	double tmp;
	if (y <= -1.9e-92) {
		tmp = t_1;
	} else if (y <= 5.2e-20) {
		tmp = x * (z / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * (z / a)
	tmp = 0
	if y <= -1.9e-92:
		tmp = t_1
	elif y <= 5.2e-20:
		tmp = x * (z / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(z / a))
	tmp = 0.0
	if (y <= -1.9e-92)
		tmp = t_1;
	elseif (y <= 5.2e-20)
		tmp = Float64(x * Float64(z / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * (z / a);
	tmp = 0.0;
	if (y <= -1.9e-92)
		tmp = t_1;
	elseif (y <= 5.2e-20)
		tmp = x * (z / t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.9e-92], t$95$1, If[LessEqual[y, 5.2e-20], N[(x * N[(z / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.9e-92 or 5.1999999999999999e-20 < y

   1. Initial program 68.1%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 48.6

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

   4. Applied rewrites 48.6%

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

   5. Taylor expanded in x 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. *-commutative N/A

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

      3. lower-*.f64 16.6

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

   7. Applied rewrites 16.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 19.2

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

   9. Applied rewrites 19.2%

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

   if -1.9e-92 < y < 5.1999999999999999e-20

   1. Initial program 68.1%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 19.5

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

   7. Applied rewrites 19.5%

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

   8. Taylor expanded in z around inf

      \[\leadsto \frac{x \cdot z}{t} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 18.6

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

   10. Applied rewrites 18.6%

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

Alternative 16: 18.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{z}{t} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return x * (z / 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 * (z / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(x * N[(z / t), $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 t around -inf

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

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

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

   7. lower-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower--.f64 46.2

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

4. Applied rewrites 46.2%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lift--.f64 19.5

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

7. Applied rewrites 19.5%

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

8. Taylor expanded in z around inf

   \[\leadsto \frac{x \cdot z}{t} \]
9. Step-by-step derivation

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 18.6

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

10. Applied rewrites 18.6%

    \[\leadsto x \cdot \frac{z}{\color{blue}{t}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025139 
(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))))