Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A

Percentage Accurate: 98.0% → 98.9%

Time: 3.7s

Alternatives: 15

Speedup: 0.6×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.3%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 99.2

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

   4. Applied rewrites 99.2%

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

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

   1. Initial program 98.8%

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

Alternative 2: 84.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \frac{\left(-t\right) \cdot y}{z - a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+99}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -6000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (/ (* (- t) y) (- z a))))
   (if (<= t_1 -5e+99)
     t_2
     (if (<= t_1 -6000000.0)
       (fma y (/ (- z t) z) x)
       (if (<= t_1 2.0)
         (fma y (/ z (- z a)) x)
         (if (<= t_1 5e+109) (fma (/ t a) y x) t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = (-t * y) / (z - a);
	double tmp;
	if (t_1 <= -5e+99) {
		tmp = t_2;
	} else if (t_1 <= -6000000.0) {
		tmp = fma(y, ((z - t) / z), x);
	} else if (t_1 <= 2.0) {
		tmp = fma(y, (z / (z - a)), x);
	} else if (t_1 <= 5e+109) {
		tmp = fma((t / a), y, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(Float64(Float64(-t) * y) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -5e+99)
		tmp = t_2;
	elseif (t_1 <= -6000000.0)
		tmp = fma(y, Float64(Float64(z - t) / z), x);
	elseif (t_1 <= 2.0)
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	elseif (t_1 <= 5e+109)
		tmp = fma(Float64(t / a), y, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-t) * y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+99], t$95$2, If[LessEqual[t$95$1, -6000000.0], N[(y * N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+109], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -5.00000000000000008e99 or 5.0000000000000001e109 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 91.9%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot y}{z - a}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lift--.f64 76.9

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

   4. Applied rewrites 76.9%

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

   if -5.00000000000000008e99 < (/.f64 (-.f64 z t) (-.f64 z a)) < -6e6

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 59.0

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

   4. Applied rewrites 59.0%

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

   if -6e6 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 99.4%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 92.1

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

   4. Applied rewrites 92.1%

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

   if 2 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5.0000000000000001e109

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 62.4

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

   4. Applied rewrites 62.4%

      \[\leadsto x + y \cdot \color{blue}{\frac{t}{a}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 62.4

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

   6. Applied rewrites 62.4%

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

Alternative 3: 84.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(-t\right) \cdot y}{z - a}\\ \mathbf{elif}\;t\_1 \leq -6000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z}, x\right)\\ \mathbf{elif}\;t\_1 \leq 50000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{z - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -5e+99)
     (/ (* (- t) y) (- z a))
     (if (<= t_1 -6000000.0)
       (fma y (/ (- z t) z) x)
       (if (<= t_1 50000000.0)
         (fma y (/ z (- z a)) x)
         (* (- z t) (/ y (- z a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -5e+99) {
		tmp = (-t * y) / (z - a);
	} else if (t_1 <= -6000000.0) {
		tmp = fma(y, ((z - t) / z), x);
	} else if (t_1 <= 50000000.0) {
		tmp = fma(y, (z / (z - a)), x);
	} else {
		tmp = (z - t) * (y / (z - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -5e+99)
		tmp = Float64(Float64(Float64(-t) * y) / Float64(z - a));
	elseif (t_1 <= -6000000.0)
		tmp = fma(y, Float64(Float64(z - t) / z), x);
	elseif (t_1 <= 50000000.0)
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	else
		tmp = Float64(Float64(z - t) * Float64(y / Float64(z - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+99], N[(N[((-t) * y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -6000000.0], N[(y * N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 50000000.0], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(z - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -5.00000000000000008e99

   1. Initial program 92.8%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot y}{z - a}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lift--.f64 76.8

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

   4. Applied rewrites 76.8%

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

   if -5.00000000000000008e99 < (/.f64 (-.f64 z t) (-.f64 z a)) < -6e6

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 59.0

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

   4. Applied rewrites 59.0%

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

   if -6e6 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e7

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 91.8

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

   4. Applied rewrites 91.8%

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

   if 5e7 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 66.0

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

   4. Applied rewrites 66.0%

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

      1. +-commutative 66.0

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

      2. *-commutative 66.0

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 68.8

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

   6. Applied rewrites 68.8%

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

Alternative 4: 84.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(-t\right) \cdot y}{z - a}\\ \mathbf{elif}\;t\_1 \leq -6000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z}, x\right)\\ \mathbf{elif}\;t\_1 \leq 50000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -5e+99)
     (/ (* (- t) y) (- z a))
     (if (<= t_1 -6000000.0)
       (fma y (/ (- z t) z) x)
       (if (<= t_1 50000000.0)
         (fma y (/ z (- z a)) x)
         (* (- t) (/ y (- z a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -5e+99) {
		tmp = (-t * y) / (z - a);
	} else if (t_1 <= -6000000.0) {
		tmp = fma(y, ((z - t) / z), x);
	} else if (t_1 <= 50000000.0) {
		tmp = fma(y, (z / (z - a)), x);
	} else {
		tmp = -t * (y / (z - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -5e+99)
		tmp = Float64(Float64(Float64(-t) * y) / Float64(z - a));
	elseif (t_1 <= -6000000.0)
		tmp = fma(y, Float64(Float64(z - t) / z), x);
	elseif (t_1 <= 50000000.0)
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	else
		tmp = Float64(Float64(-t) * Float64(y / Float64(z - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+99], N[(N[((-t) * y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -6000000.0], N[(y * N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 50000000.0], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[((-t) * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -5.00000000000000008e99

   1. Initial program 92.8%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot y}{z - a}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lift--.f64 76.8

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

   4. Applied rewrites 76.8%

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

   if -5.00000000000000008e99 < (/.f64 (-.f64 z t) (-.f64 z a)) < -6e6

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 59.0

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

   4. Applied rewrites 59.0%

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

   if -6e6 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e7

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 91.8

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

   4. Applied rewrites 91.8%

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

   if 5e7 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 66.0

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

   4. Applied rewrites 66.0%

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

      1. +-commutative 66.0

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

      2. *-commutative 66.0

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 68.8

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

   6. Applied rewrites 68.8%

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

   7. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 68.7

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

   9. Applied rewrites 68.7%

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

Alternative 5: 81.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+160}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -6000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma t (/ y a) x)))
   (if (<= t_1 -5e+160)
     t_2
     (if (<= t_1 -6000000.0)
       (fma y (/ (- z t) z) x)
       (if (<= t_1 2.0) (fma y (/ z (- z a)) x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = fma(t, (y / a), x);
	double tmp;
	if (t_1 <= -5e+160) {
		tmp = t_2;
	} else if (t_1 <= -6000000.0) {
		tmp = fma(y, ((z - t) / z), x);
	} else if (t_1 <= 2.0) {
		tmp = fma(y, (z / (z - a)), x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = fma(t, Float64(y / a), x)
	tmp = 0.0
	if (t_1 <= -5e+160)
		tmp = t_2;
	elseif (t_1 <= -6000000.0)
		tmp = fma(y, Float64(Float64(z - t) / z), x);
	elseif (t_1 <= 2.0)
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+160], t$95$2, If[LessEqual[t$95$1, -6000000.0], N[(y * N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -5.0000000000000002e160 or 2 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 93.2%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 61.5

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

   4. Applied rewrites 61.5%

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

   if -5.0000000000000002e160 < (/.f64 (-.f64 z t) (-.f64 z a)) < -6e6

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 59.4

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

   4. Applied rewrites 59.4%

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

   if -6e6 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 99.4%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 92.1

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

   4. Applied rewrites 92.1%

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

Alternative 6: 71.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \frac{t \cdot y}{a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+99}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-48}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+109}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (/ (* t y) a)))
   (if (<= t_1 -5e+99)
     t_2
     (if (<= t_1 5e-48) x (if (<= t_1 5e+109) (+ x y) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = (t * y) / a;
	double tmp;
	if (t_1 <= -5e+99) {
		tmp = t_2;
	} else if (t_1 <= 5e-48) {
		tmp = x;
	} else if (t_1 <= 5e+109) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (z - t) / (z - a)
    t_2 = (t * y) / a
    if (t_1 <= (-5d+99)) then
        tmp = t_2
    else if (t_1 <= 5d-48) then
        tmp = x
    else if (t_1 <= 5d+109) then
        tmp = x + y
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = (t * y) / a;
	double tmp;
	if (t_1 <= -5e+99) {
		tmp = t_2;
	} else if (t_1 <= 5e-48) {
		tmp = x;
	} else if (t_1 <= 5e+109) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	t_2 = (t * y) / a
	tmp = 0
	if t_1 <= -5e+99:
		tmp = t_2
	elif t_1 <= 5e-48:
		tmp = x
	elif t_1 <= 5e+109:
		tmp = x + y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(Float64(t * y) / a)
	tmp = 0.0
	if (t_1 <= -5e+99)
		tmp = t_2;
	elseif (t_1 <= 5e-48)
		tmp = x;
	elseif (t_1 <= 5e+109)
		tmp = Float64(x + y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	t_2 = (t * y) / a;
	tmp = 0.0;
	if (t_1 <= -5e+99)
		tmp = t_2;
	elseif (t_1 <= 5e-48)
		tmp = x;
	elseif (t_1 <= 5e+109)
		tmp = x + y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+99], t$95$2, If[LessEqual[t$95$1, 5e-48], x, If[LessEqual[t$95$1, 5e+109], N[(x + y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -5.00000000000000008e99 or 5.0000000000000001e109 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 91.9%

      \[x + y \cdot \frac{z - t}{z - a} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 91.9

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

   3. Applied rewrites 91.9%

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

   4. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot y}{z - a}} \]
   5. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift--.f64 76.9

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

   6. Applied rewrites 76.9%

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

   7. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 47.3

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

   9. Applied rewrites 47.3%

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

   if -5.00000000000000008e99 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.9999999999999999e-48

   1. Initial program 99.0%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 67.1%

      \[\leadsto \color{blue}{x} \]

   if 4.9999999999999999e-48 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5.0000000000000001e109

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 85.9%

      \[\leadsto x + \color{blue}{y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 80.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+99}:\\ \;\;\;\;x + \frac{t \cdot y}{a}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -5e+99)
     (+ x (/ (* t y) a))
     (if (<= t_1 2.0) (fma y (/ z (- z a)) x) (fma t (/ y a) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -5e+99) {
		tmp = x + ((t * y) / a);
	} else if (t_1 <= 2.0) {
		tmp = fma(y, (z / (z - a)), x);
	} else {
		tmp = fma(t, (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -5e+99)
		tmp = Float64(x + Float64(Float64(t * y) / a));
	elseif (t_1 <= 2.0)
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	else
		tmp = fma(t, Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+99], N[(x + N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -5.00000000000000008e99

   1. Initial program 92.8%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 60.6

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

   4. Applied rewrites 60.6%

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

   if -5.00000000000000008e99 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 87.5

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

   4. Applied rewrites 87.5%

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

   if 2 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 61.6

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

   4. Applied rewrites 61.6%

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

Alternative 8: 80.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 0.0001:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 0.0001)
     (+ x (* y (/ t a)))
     (if (<= t_1 2.0) (+ x y) (fma t (/ y a) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 0.0001) {
		tmp = x + (y * (t / a));
	} else if (t_1 <= 2.0) {
		tmp = x + y;
	} else {
		tmp = fma(t, (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 0.0001)
		tmp = Float64(x + Float64(y * Float64(t / a)));
	elseif (t_1 <= 2.0)
		tmp = Float64(x + y);
	else
		tmp = fma(t, Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0001], N[(x + N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(x + y), $MachinePrecision], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000005e-4

   1. Initial program 97.8%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 75.8

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

   4. Applied rewrites 75.8%

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

   if 1.00000000000000005e-4 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 98.4%

      \[\leadsto x + \color{blue}{y} \]

   if 2 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 61.6

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

   4. Applied rewrites 61.6%

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

Alternative 9: 80.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 0.0001)
     (fma (/ t a) y x)
     (if (<= t_1 2.0) (+ x y) (fma t (/ y a) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 0.0001) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 2.0) {
		tmp = x + y;
	} else {
		tmp = fma(t, (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 0.0001)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 2.0)
		tmp = Float64(x + y);
	else
		tmp = fma(t, Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0001], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(x + y), $MachinePrecision], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000005e-4

   1. Initial program 97.8%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 75.8

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

   4. Applied rewrites 75.8%

      \[\leadsto x + y \cdot \color{blue}{\frac{t}{a}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 75.8

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

   6. Applied rewrites 75.8%

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

   if 1.00000000000000005e-4 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 98.4%

      \[\leadsto x + \color{blue}{y} \]

   if 2 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 61.6

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

   4. Applied rewrites 61.6%

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

Alternative 10: 80.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq 0.0001:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma t (/ y a) x)))
   (if (<= t_1 0.0001) t_2 (if (<= t_1 2.0) (+ x y) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = fma(t, (y / a), x);
	double tmp;
	if (t_1 <= 0.0001) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = fma(t, Float64(y / a), x)
	tmp = 0.0
	if (t_1 <= 0.0001)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = Float64(x + y);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0001], t$95$2, If[LessEqual[t$95$1, 2.0], N[(x + y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000005e-4 or 2 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 97.1%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 72.2

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

   4. Applied rewrites 72.2%

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

   if 1.00000000000000005e-4 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 98.4%

      \[\leadsto x + \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+290}:\\ \;\;\;\;x + y \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 2e+290) (+ x (* y t_1)) (* (- t) (/ y (- z a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 2e+290) {
		tmp = x + (y * t_1);
	} else {
		tmp = -t * (y / (z - a));
	}
	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 = (z - t) / (z - a)
    if (t_1 <= 2d+290) then
        tmp = x + (y * t_1)
    else
        tmp = -t * (y / (z - a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 2e+290) {
		tmp = x + (y * t_1);
	} else {
		tmp = -t * (y / (z - a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	tmp = 0
	if t_1 <= 2e+290:
		tmp = x + (y * t_1)
	else:
		tmp = -t * (y / (z - a))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 2e+290)
		tmp = Float64(x + Float64(y * t_1));
	else
		tmp = Float64(Float64(-t) * Float64(y / Float64(z - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	tmp = 0.0;
	if (t_1 <= 2e+290)
		tmp = x + (y * t_1);
	else
		tmp = -t * (y / (z - a));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000012e290

   1. Initial program 98.8%

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

   if 2.00000000000000012e290 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 67.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 87.9

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

   4. Applied rewrites 87.9%

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

      1. +-commutative 87.9

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

      2. *-commutative 87.9

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 87.9

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

   6. Applied rewrites 87.9%

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

   7. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 87.9

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

   9. Applied rewrites 87.9%

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

Alternative 12: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+290}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 2e+290) (fma t_1 y x) (* (- t) (/ y (- z a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 2e+290) {
		tmp = fma(t_1, y, x);
	} else {
		tmp = -t * (y / (z - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 2e+290)
		tmp = fma(t_1, y, x);
	else
		tmp = Float64(Float64(-t) * Float64(y / Float64(z - a)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000012e290

   1. Initial program 98.8%

      \[x + y \cdot \frac{z - t}{z - a} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 98.8

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

   3. Applied rewrites 98.8%

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

   if 2.00000000000000012e290 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 67.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 87.9

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

   4. Applied rewrites 87.9%

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

      1. +-commutative 87.9

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

      2. *-commutative 87.9

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 87.9

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

   6. Applied rewrites 87.9%

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

   7. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 87.9

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

   9. Applied rewrites 87.9%

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

Alternative 13: 52.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \frac{z - t}{z - a} \leq 2 \cdot 10^{+205}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* y (/ (- z t) (- z a))) 2e+205) x y))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y * ((z - t) / (z - a))) <= 2e+205) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y * ((z - t) / (z - a))) <= 2d+205) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y * ((z - t) / (z - a))) <= 2e+205) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (y * ((z - t) / (z - a))) <= 2e+205:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(y * Float64(Float64(z - t) / Float64(z - a))) <= 2e+205)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((y * ((z - t) / (z - a))) <= 2e+205)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+205], x, y]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) < 2.00000000000000003e205

   1. Initial program 98.6%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 55.1%

      \[\leadsto \color{blue}{x} \]

   if 2.00000000000000003e205 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a)))

   1. Initial program 93.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 65.5

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

   4. Applied rewrites 65.5%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 45.1%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 11.2%

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

   11. Taylor expanded in z around inf

       \[\leadsto y \]
   12. Step-by-step derivation

       1. associate-*r/ 25.9

          \[\leadsto y \]

   13. Applied rewrites 25.9%

       \[\leadsto y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 66.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 5 \cdot 10^{-48}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- z t) (- z a)) 5e-48) x (+ x y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (z - a)) <= 5e-48) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((z - t) / (z - a)) <= 5d-48) then
        tmp = x
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (z - a)) <= 5e-48) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((z - t) / (z - a)) <= 5e-48:
		tmp = x
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z - t) / Float64(z - a)) <= 5e-48)
		tmp = x;
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((z - t) / (z - a)) <= 5e-48)
		tmp = x;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision], 5e-48], x, N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 4.9999999999999999e-48

   1. Initial program 97.7%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 57.4%

      \[\leadsto \color{blue}{x} \]

   if 4.9999999999999999e-48 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.4%

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

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 75.1%

      \[\leadsto x + \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 50.2% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return x;
}

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
end function

Java

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

Python

def code(x, y, z, t, a):
	return x

Julia

function code(x, y, z, t, a)
	return x
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_, a_] := x

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 50.2%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Developer Target 1: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ x + \frac{y}{\frac{z - a}{z - t}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025105 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ x (/ y (/ (- z a) (- z t)))))

  (+ x (* y (/ (- z t) (- z a)))))