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

Percentage Accurate: 98.1% → 99.2%

Time: 3.3s

Alternatives: 12

Speedup: 0.4×

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: 98.1% 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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{t \cdot y}{a - z}\\ \mathbf{elif}\;t\_1 \leq 50:\\ \;\;\;\;x + y \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(y, \frac{z}{\left(z - a\right) \cdot t}, -\frac{y}{z - a}\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 (- INFINITY))
     (/ (* t y) (- a z))
     (if (<= t_1 50.0)
       (+ x (* y t_1))
       (+ x (* (fma y (/ z (* (- z a) t)) (- (/ y (- z a)))) t))))))

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 <= -((double) INFINITY)) {
		tmp = (t * y) / (a - z);
	} else if (t_1 <= 50.0) {
		tmp = x + (y * t_1);
	} else {
		tmp = x + (fma(y, (z / ((z - a) * t)), -(y / (z - a))) * t);
	}
	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 <= Float64(-Inf))
		tmp = Float64(Float64(t * y) / Float64(a - z));
	elseif (t_1 <= 50.0)
		tmp = Float64(x + Float64(y * t_1));
	else
		tmp = Float64(x + Float64(fma(y, Float64(z / Float64(Float64(z - a) * t)), Float64(-Float64(y / Float64(z - a)))) * t));
	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, (-Infinity)], N[(N[(t * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 50.0], N[(x + N[(y * t$95$1), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y * N[(z / N[(N[(z - a), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + (-N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -inf.0

   1. Initial program 65.7%

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

   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. mul-1-neg N/A

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

      3. negate-sub2 N/A

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

      4. frac-2neg-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 90.6

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

   4. Applied rewrites 90.6%

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

   if -inf.0 < (/.f64 (-.f64 z t) (-.f64 z a)) < 50

   1. Initial program 99.6%

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

   if 50 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 95.5%

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

   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. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 94.9

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

   4. Applied rewrites 94.9%

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

Alternative 2: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \frac{t \cdot y}{a - z}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;x + y \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (/ (* t y) (- a z))))
   (if (<= t_1 (- INFINITY)) t_2 (if (<= t_1 5e+286) (+ x (* y t_1)) 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 - z);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 5e+286) {
		tmp = x + (y * t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

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 - z);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 <= 5e+286) {
		tmp = x + (y * t_1);
	} 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 - z)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_2
	elif t_1 <= 5e+286:
		tmp = x + (y * t_1)
	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) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 5e+286)
		tmp = Float64(x + Float64(y * t_1));
	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 - z);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_2;
	elseif (t_1 <= 5e+286)
		tmp = x + (y * t_1);
	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] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 5e+286], N[(x + N[(y * t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -inf.0 or 5.0000000000000004e286 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 69.5%

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

   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. mul-1-neg N/A

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

      3. negate-sub2 N/A

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

      4. frac-2neg-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 90.8

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

   4. Applied rewrites 90.8%

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

   if -inf.0 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5.0000000000000004e286

   1. Initial program 99.6%

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

Alternative 3: 82.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \frac{t \cdot y}{a - z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+44}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 10000000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (/ (* t y) (- a z))))
   (if (<= t_1 -5e+44)
     t_2
     (if (<= t_1 -2e-31)
       (fma (/ t a) y x)
       (if (<= t_1 10000000000000.0) (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 = (t * y) / (a - z);
	double tmp;
	if (t_1 <= -5e+44) {
		tmp = t_2;
	} else if (t_1 <= -2e-31) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 10000000000000.0) {
		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 = Float64(Float64(t * y) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= -5e+44)
		tmp = t_2;
	elseif (t_1 <= -2e-31)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 10000000000000.0)
		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[(N[(t * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+44], t$95$2, If[LessEqual[t$95$1, -2e-31], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 10000000000000.0], 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)) < -4.9999999999999996e44 or 1e13 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.7%

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

   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. mul-1-neg N/A

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

      3. negate-sub2 N/A

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

      4. frac-2neg-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 66.1

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

   4. Applied rewrites 66.1%

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

   if -4.9999999999999996e44 < (/.f64 (-.f64 z t) (-.f64 z a)) < -2e-31

   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 70.0

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

   4. Applied rewrites 70.0%

      \[\leadsto x + y \cdot \color{blue}{\frac{t}{a}} \]
   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 70.0

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

   6. Applied rewrites 70.0%

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

   if -2e-31 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e13

   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 91.4

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

   4. Applied rewrites 91.4%

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

Alternative 4: 81.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \frac{t \cdot y}{a - z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+44}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-87}:\\ \;\;\;\;\left(-\frac{z \cdot y}{a}\right) + x\\ \mathbf{elif}\;t\_1 \leq 10000000000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (/ (* t y) (- a z))))
   (if (<= t_1 -5e+44)
     t_2
     (if (<= t_1 -2e-31)
       (fma (/ t a) y x)
       (if (<= t_1 2e-87)
         (+ (- (/ (* z y) a)) x)
         (if (<= t_1 10000000000000.0) (+ 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 = (t * y) / (a - z);
	double tmp;
	if (t_1 <= -5e+44) {
		tmp = t_2;
	} else if (t_1 <= -2e-31) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 2e-87) {
		tmp = -((z * y) / a) + x;
	} else if (t_1 <= 10000000000000.0) {
		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 = Float64(Float64(t * y) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= -5e+44)
		tmp = t_2;
	elseif (t_1 <= -2e-31)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 2e-87)
		tmp = Float64(Float64(-Float64(Float64(z * y) / a)) + x);
	elseif (t_1 <= 10000000000000.0)
		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[(N[(t * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+44], t$95$2, If[LessEqual[t$95$1, -2e-31], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 2e-87], N[((-N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision]) + x), $MachinePrecision], If[LessEqual[t$95$1, 10000000000000.0], N[(x + y), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -4.9999999999999996e44 or 1e13 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.7%

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

   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. mul-1-neg N/A

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

      3. negate-sub2 N/A

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

      4. frac-2neg-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 66.1

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

   4. Applied rewrites 66.1%

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

   if -4.9999999999999996e44 < (/.f64 (-.f64 z t) (-.f64 z a)) < -2e-31

   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 70.0

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

   4. Applied rewrites 70.0%

      \[\leadsto x + y \cdot \color{blue}{\frac{t}{a}} \]
   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 70.0

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

   6. Applied rewrites 70.0%

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

   if -2e-31 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000004e-87

   1. Initial program 98.8%

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

   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 86.3

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

   4. Applied rewrites 86.3%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot z}{a} + x \]

      2. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot z}{a} + x \]

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 84.1

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

   7. Applied rewrites 84.1%

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

   if 2.00000000000000004e-87 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e13

   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 90.4%

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

Alternative 5: 80.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \frac{t \cdot y}{a - z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+44}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-87}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 10000000000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (/ (* t y) (- a z))))
   (if (<= t_1 -5e+44)
     t_2
     (if (<= t_1 2e-87)
       (fma (/ t a) y x)
       (if (<= t_1 10000000000000.0) (+ 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 = (t * y) / (a - z);
	double tmp;
	if (t_1 <= -5e+44) {
		tmp = t_2;
	} else if (t_1 <= 2e-87) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 10000000000000.0) {
		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 = Float64(Float64(t * y) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= -5e+44)
		tmp = t_2;
	elseif (t_1 <= 2e-87)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 10000000000000.0)
		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[(N[(t * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+44], t$95$2, If[LessEqual[t$95$1, 2e-87], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 10000000000000.0], N[(x + y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -4.9999999999999996e44 or 1e13 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.7%

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

   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. mul-1-neg N/A

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

      3. negate-sub2 N/A

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

      4. frac-2neg-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 66.1

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

   4. Applied rewrites 66.1%

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

   if -4.9999999999999996e44 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000004e-87

   1. Initial program 99.0%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 83.5

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

   4. Applied rewrites 83.5%

      \[\leadsto x + y \cdot \color{blue}{\frac{t}{a}} \]
   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 83.5

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

   6. Applied rewrites 83.5%

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

   if 2.00000000000000004e-87 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e13

   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 90.4%

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

Alternative 6: 77.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := -t \cdot \frac{y}{z}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+252}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-87}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+40}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (- (* t (/ y z)))))
   (if (<= t_1 -4e+252)
     t_2
     (if (<= t_1 2e-87) (fma (/ t a) y x) (if (<= t_1 4e+40) (+ 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 / z));
	double tmp;
	if (t_1 <= -4e+252) {
		tmp = t_2;
	} else if (t_1 <= 2e-87) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 4e+40) {
		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 * Float64(y / z)))
	tmp = 0.0
	if (t_1 <= -4e+252)
		tmp = t_2;
	elseif (t_1 <= 2e-87)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 4e+40)
		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 / z), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[t$95$1, -4e+252], t$95$2, If[LessEqual[t$95$1, 2e-87], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 4e+40], N[(x + y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -4.0000000000000004e252 or 4.00000000000000012e40 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 91.2%

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

   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 57.3

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

   4. Applied rewrites 57.3%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 44.6

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

   7. Applied rewrites 44.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 46.1

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

   9. Applied rewrites 46.1%

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

   if -4.0000000000000004e252 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000004e-87

   1. Initial program 99.2%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 78.8

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

   4. Applied rewrites 78.8%

      \[\leadsto x + y \cdot \color{blue}{\frac{t}{a}} \]
   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 78.8

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

   6. Applied rewrites 78.8%

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

   if 2.00000000000000004e-87 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.00000000000000012e40

   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 88.6%

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

Alternative 7: 77.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-87}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+40}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-t \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 2e-87)
     (fma t (/ y a) x)
     (if (<= t_1 4e+40) (+ x y) (- (* t (/ y z)))))))

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-87) {
		tmp = fma(t, (y / a), x);
	} else if (t_1 <= 4e+40) {
		tmp = x + y;
	} else {
		tmp = -(t * (y / z));
	}
	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-87)
		tmp = fma(t, Float64(y / a), x);
	elseif (t_1 <= 4e+40)
		tmp = Float64(x + y);
	else
		tmp = Float64(-Float64(t * Float64(y / z)));
	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-87], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 4e+40], N[(x + y), $MachinePrecision], (-N[(t * N[(y / z), $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000004e-87

   1. Initial program 97.5%

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

   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 77.2

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

   4. Applied rewrites 77.2%

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

   if 2.00000000000000004e-87 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.00000000000000012e40

   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 88.6%

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

   if 4.00000000000000012e40 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.6%

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

   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 59.1

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

   4. Applied rewrites 59.1%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 42.0

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

   7. Applied rewrites 42.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 43.9

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

   9. Applied rewrites 43.9%

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

Alternative 8: 71.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := -t \cdot \frac{y}{z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+36}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-87}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+40}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (- (* t (/ y z)))))
   (if (<= t_1 -5e+36)
     t_2
     (if (<= t_1 2e-87) x (if (<= t_1 4e+40) (+ 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 / z));
	double tmp;
	if (t_1 <= -5e+36) {
		tmp = t_2;
	} else if (t_1 <= 2e-87) {
		tmp = x;
	} else if (t_1 <= 4e+40) {
		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 / z))
    if (t_1 <= (-5d+36)) then
        tmp = t_2
    else if (t_1 <= 2d-87) then
        tmp = x
    else if (t_1 <= 4d+40) 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 / z));
	double tmp;
	if (t_1 <= -5e+36) {
		tmp = t_2;
	} else if (t_1 <= 2e-87) {
		tmp = x;
	} else if (t_1 <= 4e+40) {
		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 / z))
	tmp = 0
	if t_1 <= -5e+36:
		tmp = t_2
	elif t_1 <= 2e-87:
		tmp = x
	elif t_1 <= 4e+40:
		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 * Float64(y / z)))
	tmp = 0.0
	if (t_1 <= -5e+36)
		tmp = t_2;
	elseif (t_1 <= 2e-87)
		tmp = x;
	elseif (t_1 <= 4e+40)
		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 / z));
	tmp = 0.0;
	if (t_1 <= -5e+36)
		tmp = t_2;
	elseif (t_1 <= 2e-87)
		tmp = x;
	elseif (t_1 <= 4e+40)
		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[(t * N[(y / z), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[t$95$1, -5e+36], t$95$2, If[LessEqual[t$95$1, 2e-87], x, If[LessEqual[t$95$1, 4e+40], N[(x + y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -4.99999999999999977e36 or 4.00000000000000012e40 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.5%

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

   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 58.9

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

   4. Applied rewrites 58.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 40.9

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

   7. Applied rewrites 40.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 42.4

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

   9. Applied rewrites 42.4%

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

   if -4.99999999999999977e36 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000004e-87

   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 70.9%

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

   if 2.00000000000000004e-87 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.00000000000000012e40

   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 88.6%

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

Alternative 9: 70.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \frac{y}{a} \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+49}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-87}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+130}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* (/ y a) t)))
   (if (<= t_1 -5e+49)
     t_2
     (if (<= t_1 2e-87) x (if (<= t_1 5e+130) (+ 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 = (y / a) * t;
	double tmp;
	if (t_1 <= -5e+49) {
		tmp = t_2;
	} else if (t_1 <= 2e-87) {
		tmp = x;
	} else if (t_1 <= 5e+130) {
		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 = (y / a) * t
    if (t_1 <= (-5d+49)) then
        tmp = t_2
    else if (t_1 <= 2d-87) then
        tmp = x
    else if (t_1 <= 5d+130) 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 = (y / a) * t;
	double tmp;
	if (t_1 <= -5e+49) {
		tmp = t_2;
	} else if (t_1 <= 2e-87) {
		tmp = x;
	} else if (t_1 <= 5e+130) {
		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 = (y / a) * t
	tmp = 0
	if t_1 <= -5e+49:
		tmp = t_2
	elif t_1 <= 2e-87:
		tmp = x
	elif t_1 <= 5e+130:
		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(y / a) * t)
	tmp = 0.0
	if (t_1 <= -5e+49)
		tmp = t_2;
	elseif (t_1 <= 2e-87)
		tmp = x;
	elseif (t_1 <= 5e+130)
		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 = (y / a) * t;
	tmp = 0.0;
	if (t_1 <= -5e+49)
		tmp = t_2;
	elseif (t_1 <= 2e-87)
		tmp = x;
	elseif (t_1 <= 5e+130)
		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[(y / a), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+49], t$95$2, If[LessEqual[t$95$1, 2e-87], x, If[LessEqual[t$95$1, 5e+130], N[(x + y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -5.0000000000000004e49 or 4.9999999999999996e130 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 92.8%

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

   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 64.8

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

   4. Applied rewrites 64.8%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 48.2

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

   7. Applied rewrites 48.2%

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

   if -5.0000000000000004e49 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000004e-87

   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 70.0%

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

   if 2.00000000000000004e-87 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.9999999999999996e130

   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 82.8%

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

Alternative 10: 66.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 5 \cdot 10^{-87}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- z t) (- z a)) 5e-87) x (+ x y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (z - a)) <= 5e-87) {
		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-87) 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-87) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((z - t) / (z - a)) <= 5e-87:
		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-87)
		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-87)
		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-87], x, N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 5.00000000000000042e-87

   1. Initial program 97.5%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 57.9%

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

   if 5.00000000000000042e-87 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.6%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 73.8%

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

Alternative 11: 51.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 0.9999999999618374:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 1.000000001:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 0.9999999999618374) x (if (<= t_1 1.000000001) y 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 <= 0.9999999999618374) {
		tmp = x;
	} else if (t_1 <= 1.000000001) {
		tmp = y;
	} else {
		tmp = x;
	}
	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 <= 0.9999999999618374d0) then
        tmp = x
    else if (t_1 <= 1.000000001d0) then
        tmp = y
    else
        tmp = x
    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 <= 0.9999999999618374) {
		tmp = x;
	} else if (t_1 <= 1.000000001) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	tmp = 0
	if t_1 <= 0.9999999999618374:
		tmp = x
	elif t_1 <= 1.000000001:
		tmp = y
	else:
		tmp = 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 <= 0.9999999999618374)
		tmp = x;
	elseif (t_1 <= 1.000000001)
		tmp = y;
	else
		tmp = x;
	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 <= 0.9999999999618374)
		tmp = x;
	elseif (t_1 <= 1.000000001)
		tmp = y;
	else
		tmp = x;
	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, 0.9999999999618374], x, If[LessEqual[t$95$1, 1.000000001], y, x]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 0.99999999996183742 or 1.0000000010000001 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 97.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 50.8%

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

   if 0.99999999996183742 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.0000000010000001

   1. Initial program 100.0%

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

   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. negate-sub N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. negate-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 49.4

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

   7. Applied rewrites 49.4%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 49.3%

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

Alternative 12: 50.4% accurate, 15.3× 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 98.1%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 51.2%

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

Reproduce

herbie shell --seed 2025110 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64
  (+ x (* y (/ (- z t) (- z a)))))