Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3

Percentage Accurate: 84.0% → 97.1%

Time: 3.1s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift--.f64 97.1

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

3. Applied rewrites 97.1%

   \[\leadsto \color{blue}{x \cdot \frac{y - z}{t - z}} \]
4. Add Preprocessing

Alternative 2: 75.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y - z}{t}\\ t_2 := \mathsf{fma}\left(-x, \frac{y}{z}, x\right)\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{+36}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-36}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+49}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ (- y z) t))) (t_2 (fma (- x) (/ y z) x)))
   (if (<= z -2.7e+36)
     t_2
     (if (<= z -1.3e-43)
       t_1
       (if (<= z 6e-36) (* x (/ y (- t z))) (if (<= z 2.1e+49) t_1 t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((y - z) / t);
	double t_2 = fma(-x, (y / z), x);
	double tmp;
	if (z <= -2.7e+36) {
		tmp = t_2;
	} else if (z <= -1.3e-43) {
		tmp = t_1;
	} else if (z <= 6e-36) {
		tmp = x * (y / (t - z));
	} else if (z <= 2.1e+49) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(y - z) / t))
	t_2 = fma(Float64(-x), Float64(y / z), x)
	tmp = 0.0
	if (z <= -2.7e+36)
		tmp = t_2;
	elseif (z <= -1.3e-43)
		tmp = t_1;
	elseif (z <= 6e-36)
		tmp = Float64(x * Float64(y / Float64(t - z)));
	elseif (z <= 2.1e+49)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-x) * N[(y / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -2.7e+36], t$95$2, If[LessEqual[z, -1.3e-43], t$95$1, If[LessEqual[z, 6e-36], N[(x * N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.1e+49], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.7000000000000001e36 or 2.10000000000000011e49 < z

   1. Initial program 72.2%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

      5. distribute-lft-out-- N/A

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 68.2

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

   4. Applied rewrites 68.2%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 3.9

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

   7. Applied rewrites 3.9%

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

   8. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-/.f64 78.4

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

   10. Applied rewrites 78.4%

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

   if -2.7000000000000001e36 < z < -1.3e-43 or 6.0000000000000003e-36 < z < 2.10000000000000011e49

   1. Initial program 95.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift--.f64 99.2

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 52.0%

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

   if -1.3e-43 < z < 6.0000000000000003e-36

   1. Initial program 92.4%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 78.9

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

   4. Applied rewrites 78.9%

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

Alternative 3: 75.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-x, \frac{y}{z}, x\right)\\ \mathbf{if}\;z \leq -2 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+77}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (- x) (/ y z) x)))
   (if (<= z -2e+24) t_1 (if (<= z 3.1e+77) (* x (/ y (- t z))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(-x, (y / z), x);
	double tmp;
	if (z <= -2e+24) {
		tmp = t_1;
	} else if (z <= 3.1e+77) {
		tmp = x * (y / (t - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(-x), Float64(y / z), x)
	tmp = 0.0
	if (z <= -2e+24)
		tmp = t_1;
	elseif (z <= 3.1e+77)
		tmp = Float64(x * Float64(y / Float64(t - z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-x) * N[(y / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -2e+24], t$95$1, If[LessEqual[z, 3.1e+77], N[(x * N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e24 or 3.09999999999999999e77 < z

   1. Initial program 71.8%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

      5. distribute-lft-out-- N/A

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 68.9

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

   4. Applied rewrites 68.9%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 3.9

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

   7. Applied rewrites 3.9%

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

   8. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-/.f64 79.2

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

   10. Applied rewrites 79.2%

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

   if -2e24 < z < 3.09999999999999999e77

   1. Initial program 92.8%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 72.5

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

   4. Applied rewrites 72.5%

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

Alternative 4: 65.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y}{t}\\ \mathbf{if}\;t \leq -9.6 \cdot 10^{+117}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ y t))))
   (if (<= t -9.6e+117) t_1 (if (<= t 1.85e+17) (fma (- x) (/ y z) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (y / t);
	double tmp;
	if (t <= -9.6e+117) {
		tmp = t_1;
	} else if (t <= 1.85e+17) {
		tmp = fma(-x, (y / z), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(y / t))
	tmp = 0.0
	if (t <= -9.6e+117)
		tmp = t_1;
	elseif (t <= 1.85e+17)
		tmp = fma(Float64(-x), Float64(y / z), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(y / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -9.6e+117], t$95$1, If[LessEqual[t, 1.85e+17], N[((-x) * N[(y / z), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -9.5999999999999996e117 or 1.85e17 < t

   1. Initial program 81.9%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift--.f64 96.7

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

   3. Applied rewrites 96.7%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 55.6

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

   6. Applied rewrites 55.6%

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

   if -9.5999999999999996e117 < t < 1.85e17

   1. Initial program 85.4%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

      5. distribute-lft-out-- N/A

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 67.4

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

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 5.0

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

   7. Applied rewrites 5.0%

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

   8. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-/.f64 71.1

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

   10. Applied rewrites 71.1%

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

Alternative 5: 61.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{z}, x\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+48}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.1e+26) (fma t (/ x z) x) (if (<= z 8.5e+48) (* x (/ y t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.1e+26) {
		tmp = fma(t, (x / z), x);
	} else if (z <= 8.5e+48) {
		tmp = x * (y / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.1e+26)
		tmp = fma(t, Float64(x / z), x);
	elseif (z <= 8.5e+48)
		tmp = Float64(x * Float64(y / t));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.1e+26], N[(t * N[(x / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 8.5e+48], N[(x * N[(y / t), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.10000000000000004e26

   1. Initial program 72.9%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

      5. distribute-lft-out-- N/A

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 67.9

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

   4. Applied rewrites 67.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 61.0

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

   7. Applied rewrites 61.0%

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

   if -1.10000000000000004e26 < z < 8.5000000000000001e48

   1. Initial program 93.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift--.f64 94.9

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

   3. Applied rewrites 94.9%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 61.1

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

   6. Applied rewrites 61.1%

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

   if 8.5000000000000001e48 < z

   1. Initial program 72.1%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 62.8%

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

Alternative 6: 61.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+26}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+48}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.1e+26) x (if (<= z 8.5e+48) (* x (/ y t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.1e+26) {
		tmp = x;
	} else if (z <= 8.5e+48) {
		tmp = x * (y / t);
	} 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)
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) :: tmp
    if (z <= (-1.1d+26)) then
        tmp = x
    else if (z <= 8.5d+48) then
        tmp = x * (y / t)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.1e+26) {
		tmp = x;
	} else if (z <= 8.5e+48) {
		tmp = x * (y / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.1e+26:
		tmp = x
	elif z <= 8.5e+48:
		tmp = x * (y / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.1e+26)
		tmp = x;
	elseif (z <= 8.5e+48)
		tmp = Float64(x * Float64(y / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.1e+26)
		tmp = x;
	elseif (z <= 8.5e+48)
		tmp = x * (y / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.1e+26], x, If[LessEqual[z, 8.5e+48], N[(x * N[(y / t), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -1.10000000000000004e26 or 8.5000000000000001e48 < z

   1. Initial program 72.5%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 61.9%

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

   if -1.10000000000000004e26 < z < 8.5000000000000001e48

   1. Initial program 93.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift--.f64 94.9

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

   3. Applied rewrites 94.9%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 61.1

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

   6. Applied rewrites 61.1%

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

Alternative 7: 60.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.08 \cdot 10^{+26}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.08e+26) x (if (<= z 8.5e+48) (/ (* y x) t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.08e+26) {
		tmp = x;
	} else if (z <= 8.5e+48) {
		tmp = (y * x) / t;
	} 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)
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) :: tmp
    if (z <= (-1.08d+26)) then
        tmp = x
    else if (z <= 8.5d+48) then
        tmp = (y * x) / t
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.08e+26) {
		tmp = x;
	} else if (z <= 8.5e+48) {
		tmp = (y * x) / t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.08e+26:
		tmp = x
	elif z <= 8.5e+48:
		tmp = (y * x) / t
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.08e+26)
		tmp = x;
	elseif (z <= 8.5e+48)
		tmp = Float64(Float64(y * x) / t);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.08e+26)
		tmp = x;
	elseif (z <= 8.5e+48)
		tmp = (y * x) / t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.08e+26], x, If[LessEqual[z, 8.5e+48], N[(N[(y * x), $MachinePrecision] / t), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -1.08e26 or 8.5000000000000001e48 < z

   1. Initial program 72.5%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 61.9%

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

   if -1.08e26 < z < 8.5000000000000001e48

   1. Initial program 93.0%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 58.9

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

   4. Applied rewrites 58.9%

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

Alternative 8: 35.3% accurate, 23.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

2. Taylor expanded in z around inf

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

4. Applied rewrites 35.3%

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

Developer Target 1: 97.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025095 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

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

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