Complex division, imag part

Percentage Accurate: 62.4% → 83.1%

Time: 3.9s

Alternatives: 9

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))

C

double code(double a, double b, double c, double d) {
	return ((b * c) - (a * d)) / ((c * c) + (d * d));
}

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(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = ((b * c) - (a * d)) / ((c * c) + (d * d))
end function

Java

public static double code(double a, double b, double c, double d) {
	return ((b * c) - (a * d)) / ((c * c) + (d * d));
}

Python

def code(a, b, c, d):
	return ((b * c) - (a * d)) / ((c * c) + (d * d))

Julia

function code(a, b, c, d)
	return Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 62.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))

C

double code(double a, double b, double c, double d) {
	return ((b * c) - (a * d)) / ((c * c) + (d * d));
}

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(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = ((b * c) - (a * d)) / ((c * c) + (d * d))
end function

Java

public static double code(double a, double b, double c, double d) {
	return ((b * c) - (a * d)) / ((c * c) + (d * d));
}

Python

def code(a, b, c, d):
	return ((b * c) - (a * d)) / ((c * c) + (d * d))

Julia

function code(a, b, c, d)
	return Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 83.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -4.7 \cdot 10^{+72}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq -1.3 \cdot 10^{-122}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 5.7 \cdot 10^{-105}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{c}\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{+60}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{d} \cdot b - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))))
   (if (<= d -4.7e+72)
     (/ (- (* c (/ b d)) a) d)
     (if (<= d -1.3e-122)
       t_0
       (if (<= d 5.7e-105)
         (- (/ (fma a (/ d c) (- b)) c))
         (if (<= d 1.02e+60) t_0 (/ (- (* (/ c d) b) a) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -4.7e+72) {
		tmp = ((c * (b / d)) - a) / d;
	} else if (d <= -1.3e-122) {
		tmp = t_0;
	} else if (d <= 5.7e-105) {
		tmp = -(fma(a, (d / c), -b) / c);
	} else if (d <= 1.02e+60) {
		tmp = t_0;
	} else {
		tmp = (((c / d) * b) - a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -4.7e+72)
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	elseif (d <= -1.3e-122)
		tmp = t_0;
	elseif (d <= 5.7e-105)
		tmp = Float64(-Float64(fma(a, Float64(d / c), Float64(-b)) / c));
	elseif (d <= 1.02e+60)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(c / d) * b) - a) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4.7e+72], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -1.3e-122], t$95$0, If[LessEqual[d, 5.7e-105], (-N[(N[(a * N[(d / c), $MachinePrecision] + (-b)), $MachinePrecision] / c), $MachinePrecision]), If[LessEqual[d, 1.02e+60], t$95$0, N[(N[(N[(N[(c / d), $MachinePrecision] * b), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -4.70000000000000034e72

   1. Initial program 43.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a + \frac{b \cdot c}{d}}{\color{blue}{d}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} + -1 \cdot a}{d} \]

      3. associate-/l* N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} + -1 \cdot a}{d} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, \mathsf{neg}\left(a\right)\right)}{d} \]

      7. lower-neg.f64 81.0

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d} \]

   4. Applied rewrites 81.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} - a}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      5. lift-/.f64 81.0

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

   7. Applied rewrites 81.0%

      \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} - a}{d} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

      7. lift-*.f64 76.7

         \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

   9. Applied rewrites 76.7%

      \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

       3. associate-/l* N/A

          \[\leadsto \frac{c \cdot \frac{b}{d} - a}{d} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{c \cdot \frac{b}{d} - a}{d} \]

       5. lower-/.f64 82.1

          \[\leadsto \frac{c \cdot \frac{b}{d} - a}{d} \]

   11. Applied rewrites 82.1%

       \[\leadsto \frac{c \cdot \frac{b}{d} - a}{d} \]

   if -4.70000000000000034e72 < d < -1.29999999999999988e-122 or 5.69999999999999963e-105 < d < 1.0200000000000001e60

   1. Initial program 77.2%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   if -1.29999999999999988e-122 < d < 5.69999999999999963e-105

   1. Initial program 72.2%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot b + \frac{a \cdot d}{c}}{c}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot b + \frac{a \cdot d}{c}}{c}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{-1 \cdot b + \frac{a \cdot d}{c}}{c} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{-1 \cdot b + \frac{a \cdot d}{c}}{c} \]

      4. +-commutative N/A

         \[\leadsto -\frac{\frac{a \cdot d}{c} + -1 \cdot b}{c} \]

      5. associate-/l* N/A

         \[\leadsto -\frac{a \cdot \frac{d}{c} + -1 \cdot b}{c} \]

      6. lower-fma.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(a, \frac{d}{c}, -1 \cdot b\right)}{c} \]

      7. lower-/.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(a, \frac{d}{c}, -1 \cdot b\right)}{c} \]

      8. mul-1-neg N/A

         \[\leadsto -\frac{\mathsf{fma}\left(a, \frac{d}{c}, \mathsf{neg}\left(b\right)\right)}{c} \]

      9. lower-neg.f64 90.3

         \[\leadsto -\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{c} \]

   4. Applied rewrites 90.3%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{c}} \]

   if 1.0200000000000001e60 < d

   1. Initial program 44.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a + \frac{b \cdot c}{d}}{\color{blue}{d}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} + -1 \cdot a}{d} \]

      3. associate-/l* N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} + -1 \cdot a}{d} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, \mathsf{neg}\left(a\right)\right)}{d} \]

      7. lower-neg.f64 80.9

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d} \]

   4. Applied rewrites 80.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} - a}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      5. lift-/.f64 80.9

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

   7. Applied rewrites 80.9%

      \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.05 \cdot 10^{+75}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{-34}:\\ \;\;\;\;\frac{-d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a\\ \mathbf{elif}\;d \leq 1.04 \cdot 10^{+33}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{d} \cdot b - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.05e+75)
   (/ (- (* c (/ b d)) a) d)
   (if (<= d -1.4e-34)
     (* (/ (- d) (fma d d (* c c))) a)
     (if (<= d 1.04e+33)
       (- (/ (fma a (/ d c) (- b)) c))
       (/ (- (* (/ c d) b) a) d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.05e+75) {
		tmp = ((c * (b / d)) - a) / d;
	} else if (d <= -1.4e-34) {
		tmp = (-d / fma(d, d, (c * c))) * a;
	} else if (d <= 1.04e+33) {
		tmp = -(fma(a, (d / c), -b) / c);
	} else {
		tmp = (((c / d) * b) - a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.05e+75)
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	elseif (d <= -1.4e-34)
		tmp = Float64(Float64(Float64(-d) / fma(d, d, Float64(c * c))) * a);
	elseif (d <= 1.04e+33)
		tmp = Float64(-Float64(fma(a, Float64(d / c), Float64(-b)) / c));
	else
		tmp = Float64(Float64(Float64(Float64(c / d) * b) - a) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.05e+75], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -1.4e-34], N[(N[((-d) / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[d, 1.04e+33], (-N[(N[(a * N[(d / c), $MachinePrecision] + (-b)), $MachinePrecision] / c), $MachinePrecision]), N[(N[(N[(N[(c / d), $MachinePrecision] * b), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.04999999999999999e75

   1. Initial program 42.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a + \frac{b \cdot c}{d}}{\color{blue}{d}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} + -1 \cdot a}{d} \]

      3. associate-/l* N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} + -1 \cdot a}{d} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, \mathsf{neg}\left(a\right)\right)}{d} \]

      7. lower-neg.f64 81.1

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d} \]

   4. Applied rewrites 81.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} - a}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      5. lift-/.f64 81.1

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

   7. Applied rewrites 81.1%

      \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} - a}{d} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

      7. lift-*.f64 76.8

         \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

   9. Applied rewrites 76.8%

      \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

       3. associate-/l* N/A

          \[\leadsto \frac{c \cdot \frac{b}{d} - a}{d} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{c \cdot \frac{b}{d} - a}{d} \]

       5. lower-/.f64 82.3

          \[\leadsto \frac{c \cdot \frac{b}{d} - a}{d} \]

   11. Applied rewrites 82.3%

       \[\leadsto \frac{c \cdot \frac{b}{d} - a}{d} \]

   if -1.04999999999999999e75 < d < -1.39999999999999998e-34

   1. Initial program 74.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \frac{d}{{c}^{2} + {d}^{2}} + \frac{b \cdot c}{a \cdot \left({c}^{2} + {d}^{2}\right)}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{d}{{c}^{2} + {d}^{2}} + \frac{b \cdot c}{a \cdot \left({c}^{2} + {d}^{2}\right)}\right) \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \frac{d}{{c}^{2} + {d}^{2}} + \frac{b \cdot c}{a \cdot \left({c}^{2} + {d}^{2}\right)}\right) \cdot \color{blue}{a} \]

   4. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{-1 \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(d\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a \]

      2. lift-neg.f64 54.2

         \[\leadsto \frac{-d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a \]

   7. Applied rewrites 54.2%

      \[\leadsto \frac{-d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a \]

   if -1.39999999999999998e-34 < d < 1.0400000000000001e33

   1. Initial program 74.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot b + \frac{a \cdot d}{c}}{c}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot b + \frac{a \cdot d}{c}}{c}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{-1 \cdot b + \frac{a \cdot d}{c}}{c} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{-1 \cdot b + \frac{a \cdot d}{c}}{c} \]

      4. +-commutative N/A

         \[\leadsto -\frac{\frac{a \cdot d}{c} + -1 \cdot b}{c} \]

      5. associate-/l* N/A

         \[\leadsto -\frac{a \cdot \frac{d}{c} + -1 \cdot b}{c} \]

      6. lower-fma.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(a, \frac{d}{c}, -1 \cdot b\right)}{c} \]

      7. lower-/.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(a, \frac{d}{c}, -1 \cdot b\right)}{c} \]

      8. mul-1-neg N/A

         \[\leadsto -\frac{\mathsf{fma}\left(a, \frac{d}{c}, \mathsf{neg}\left(b\right)\right)}{c} \]

      9. lower-neg.f64 80.1

         \[\leadsto -\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{c} \]

   4. Applied rewrites 80.1%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{c}} \]

   if 1.0400000000000001e33 < d

   1. Initial program 46.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a + \frac{b \cdot c}{d}}{\color{blue}{d}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} + -1 \cdot a}{d} \]

      3. associate-/l* N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} + -1 \cdot a}{d} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, \mathsf{neg}\left(a\right)\right)}{d} \]

      7. lower-neg.f64 79.1

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d} \]

   4. Applied rewrites 79.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} - a}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      5. lift-/.f64 79.1

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

   7. Applied rewrites 79.1%

      \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 77.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq 1.04 \cdot 10^{+33}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{d} \cdot b - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -5.2e-51)
   (/ (- (* c (/ b d)) a) d)
   (if (<= d 1.04e+33)
     (- (/ (fma a (/ d c) (- b)) c))
     (/ (- (* (/ c d) b) a) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -5.2e-51) {
		tmp = ((c * (b / d)) - a) / d;
	} else if (d <= 1.04e+33) {
		tmp = -(fma(a, (d / c), -b) / c);
	} else {
		tmp = (((c / d) * b) - a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -5.2e-51)
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	elseif (d <= 1.04e+33)
		tmp = Float64(-Float64(fma(a, Float64(d / c), Float64(-b)) / c));
	else
		tmp = Float64(Float64(Float64(Float64(c / d) * b) - a) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -5.2e-51], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 1.04e+33], (-N[(N[(a * N[(d / c), $MachinePrecision] + (-b)), $MachinePrecision] / c), $MachinePrecision]), N[(N[(N[(N[(c / d), $MachinePrecision] * b), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.2e-51

   1. Initial program 53.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a + \frac{b \cdot c}{d}}{\color{blue}{d}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} + -1 \cdot a}{d} \]

      3. associate-/l* N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} + -1 \cdot a}{d} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, \mathsf{neg}\left(a\right)\right)}{d} \]

      7. lower-neg.f64 71.1

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d} \]

   4. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} - a}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      5. lift-/.f64 71.1

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

   7. Applied rewrites 71.1%

      \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} - a}{d} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

      7. lift-*.f64 68.3

         \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

   9. Applied rewrites 68.3%

      \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

       3. associate-/l* N/A

          \[\leadsto \frac{c \cdot \frac{b}{d} - a}{d} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{c \cdot \frac{b}{d} - a}{d} \]

       5. lower-/.f64 71.8

          \[\leadsto \frac{c \cdot \frac{b}{d} - a}{d} \]

   11. Applied rewrites 71.8%

       \[\leadsto \frac{c \cdot \frac{b}{d} - a}{d} \]

   if -5.2e-51 < d < 1.0400000000000001e33

   1. Initial program 74.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot b + \frac{a \cdot d}{c}}{c}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot b + \frac{a \cdot d}{c}}{c}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{-1 \cdot b + \frac{a \cdot d}{c}}{c} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{-1 \cdot b + \frac{a \cdot d}{c}}{c} \]

      4. +-commutative N/A

         \[\leadsto -\frac{\frac{a \cdot d}{c} + -1 \cdot b}{c} \]

      5. associate-/l* N/A

         \[\leadsto -\frac{a \cdot \frac{d}{c} + -1 \cdot b}{c} \]

      6. lower-fma.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(a, \frac{d}{c}, -1 \cdot b\right)}{c} \]

      7. lower-/.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(a, \frac{d}{c}, -1 \cdot b\right)}{c} \]

      8. mul-1-neg N/A

         \[\leadsto -\frac{\mathsf{fma}\left(a, \frac{d}{c}, \mathsf{neg}\left(b\right)\right)}{c} \]

      9. lower-neg.f64 80.5

         \[\leadsto -\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{c} \]

   4. Applied rewrites 80.5%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{c}} \]

   if 1.0400000000000001e33 < d

   1. Initial program 46.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a + \frac{b \cdot c}{d}}{\color{blue}{d}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} + -1 \cdot a}{d} \]

      3. associate-/l* N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} + -1 \cdot a}{d} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, \mathsf{neg}\left(a\right)\right)}{d} \]

      7. lower-neg.f64 79.1

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d} \]

   4. Applied rewrites 79.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} - a}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      5. lift-/.f64 79.1

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

   7. Applied rewrites 79.1%

      \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 77.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq 1.04 \cdot 10^{+33}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{d} \cdot b - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -5.2e-51)
   (/ (- (* c (/ b d)) a) d)
   (if (<= d 1.04e+33) (/ (- b (/ (* d a) c)) c) (/ (- (* (/ c d) b) a) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -5.2e-51) {
		tmp = ((c * (b / d)) - a) / d;
	} else if (d <= 1.04e+33) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = (((c / d) * b) - a) / d;
	}
	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(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-5.2d-51)) then
        tmp = ((c * (b / d)) - a) / d
    else if (d <= 1.04d+33) then
        tmp = (b - ((d * a) / c)) / c
    else
        tmp = (((c / d) * b) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -5.2e-51) {
		tmp = ((c * (b / d)) - a) / d;
	} else if (d <= 1.04e+33) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = (((c / d) * b) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -5.2e-51:
		tmp = ((c * (b / d)) - a) / d
	elif d <= 1.04e+33:
		tmp = (b - ((d * a) / c)) / c
	else:
		tmp = (((c / d) * b) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -5.2e-51)
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	elseif (d <= 1.04e+33)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	else
		tmp = Float64(Float64(Float64(Float64(c / d) * b) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -5.2e-51)
		tmp = ((c * (b / d)) - a) / d;
	elseif (d <= 1.04e+33)
		tmp = (b - ((d * a) / c)) / c;
	else
		tmp = (((c / d) * b) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -5.2e-51], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 1.04e+33], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(N[(c / d), $MachinePrecision] * b), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.2e-51

   1. Initial program 53.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a + \frac{b \cdot c}{d}}{\color{blue}{d}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} + -1 \cdot a}{d} \]

      3. associate-/l* N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} + -1 \cdot a}{d} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, \mathsf{neg}\left(a\right)\right)}{d} \]

      7. lower-neg.f64 71.1

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d} \]

   4. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} - a}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      5. lift-/.f64 71.1

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

   7. Applied rewrites 71.1%

      \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} - a}{d} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

      7. lift-*.f64 68.3

         \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

   9. Applied rewrites 68.3%

      \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

       3. associate-/l* N/A

          \[\leadsto \frac{c \cdot \frac{b}{d} - a}{d} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{c \cdot \frac{b}{d} - a}{d} \]

       5. lower-/.f64 71.8

          \[\leadsto \frac{c \cdot \frac{b}{d} - a}{d} \]

   11. Applied rewrites 71.8%

       \[\leadsto \frac{c \cdot \frac{b}{d} - a}{d} \]

   if -5.2e-51 < d < 1.0400000000000001e33

   1. Initial program 74.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{b + -1 \cdot \frac{a \cdot d}{c}}{\color{blue}{c}} \]

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

         \[\leadsto \frac{b - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot d}{c}}{c} \]

      3. metadata-eval N/A

         \[\leadsto \frac{b - 1 \cdot \frac{a \cdot d}{c}}{c} \]

      4. metadata-eval N/A

         \[\leadsto \frac{b - \frac{-1}{-1} \cdot \frac{a \cdot d}{c}}{c} \]

      5. times-frac N/A

         \[\leadsto \frac{b - \frac{-1 \cdot \left(a \cdot d\right)}{-1 \cdot c}}{c} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{b - \frac{\mathsf{neg}\left(a \cdot d\right)}{-1 \cdot c}}{c} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{b - \frac{\mathsf{neg}\left(a \cdot d\right)}{\mathsf{neg}\left(c\right)}}{c} \]

      8. frac-2neg N/A

         \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      11. *-commutative N/A

          \[\leadsto \frac{b - \frac{d \cdot a}{c}}{c} \]

      12. lower-*.f64 80.5

          \[\leadsto \frac{b - \frac{d \cdot a}{c}}{c} \]

   4. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\frac{b - \frac{d \cdot a}{c}}{c}} \]

   if 1.0400000000000001e33 < d

   1. Initial program 46.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a + \frac{b \cdot c}{d}}{\color{blue}{d}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} + -1 \cdot a}{d} \]

      3. associate-/l* N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} + -1 \cdot a}{d} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, \mathsf{neg}\left(a\right)\right)}{d} \]

      7. lower-neg.f64 79.1

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d} \]

   4. Applied rewrites 79.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} - a}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      5. lift-/.f64 79.1

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

   7. Applied rewrites 79.1%

      \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.11 \cdot 10^{+74}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.66 \cdot 10^{+75}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.11e+74)
   (/ b c)
   (if (<= c 1.66e+75) (/ (- (/ (* c b) d) a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.11e+74) {
		tmp = b / c;
	} else if (c <= 1.66e+75) {
		tmp = (((c * b) / d) - a) / d;
	} else {
		tmp = b / c;
	}
	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(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.11d+74)) then
        tmp = b / c
    else if (c <= 1.66d+75) then
        tmp = (((c * b) / d) - a) / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.11e+74) {
		tmp = b / c;
	} else if (c <= 1.66e+75) {
		tmp = (((c * b) / d) - a) / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.11e+74:
		tmp = b / c
	elif c <= 1.66e+75:
		tmp = (((c * b) / d) - a) / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.11e+74)
		tmp = Float64(b / c);
	elseif (c <= 1.66e+75)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.11e+74)
		tmp = b / c;
	elseif (c <= 1.66e+75)
		tmp = (((c * b) / d) - a) / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.11e+74], N[(b / c), $MachinePrecision], If[LessEqual[c, 1.66e+75], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.11e74 or 1.66e75 < c

   1. Initial program 42.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b}{c}} \]
   3. Step-by-step derivation

      1. lower-/.f64 70.9

         \[\leadsto \frac{b}{\color{blue}{c}} \]

   4. Applied rewrites 70.9%

      \[\leadsto \color{blue}{\frac{b}{c}} \]

   if -1.11e74 < c < 1.66e75

   1. Initial program 74.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a + \frac{b \cdot c}{d}}{\color{blue}{d}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} + -1 \cdot a}{d} \]

      3. associate-/l* N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} + -1 \cdot a}{d} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, \mathsf{neg}\left(a\right)\right)}{d} \]

      7. lower-neg.f64 73.6

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d} \]

   4. Applied rewrites 73.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} - a}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      5. lift-/.f64 73.6

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

   7. Applied rewrites 73.6%

      \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} - a}{d} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

      7. lift-*.f64 73.5

         \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

   9. Applied rewrites 73.5%

      \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 72.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+74}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.66 \cdot 10^{+75}:\\ \;\;\;\;\frac{\frac{c}{d} \cdot b - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.25e+74)
   (/ b c)
   (if (<= c 1.66e+75) (/ (- (* (/ c d) b) a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.25e+74) {
		tmp = b / c;
	} else if (c <= 1.66e+75) {
		tmp = (((c / d) * b) - a) / d;
	} else {
		tmp = b / c;
	}
	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(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.25d+74)) then
        tmp = b / c
    else if (c <= 1.66d+75) then
        tmp = (((c / d) * b) - a) / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.25e+74) {
		tmp = b / c;
	} else if (c <= 1.66e+75) {
		tmp = (((c / d) * b) - a) / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.25e+74:
		tmp = b / c
	elif c <= 1.66e+75:
		tmp = (((c / d) * b) - a) / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.25e+74)
		tmp = Float64(b / c);
	elseif (c <= 1.66e+75)
		tmp = Float64(Float64(Float64(Float64(c / d) * b) - a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.25e+74)
		tmp = b / c;
	elseif (c <= 1.66e+75)
		tmp = (((c / d) * b) - a) / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.25e+74], N[(b / c), $MachinePrecision], If[LessEqual[c, 1.66e+75], N[(N[(N[(N[(c / d), $MachinePrecision] * b), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.24999999999999991e74 or 1.66e75 < c

   1. Initial program 42.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b}{c}} \]
   3. Step-by-step derivation

      1. lower-/.f64 70.9

         \[\leadsto \frac{b}{\color{blue}{c}} \]

   4. Applied rewrites 70.9%

      \[\leadsto \color{blue}{\frac{b}{c}} \]

   if -1.24999999999999991e74 < c < 1.66e75

   1. Initial program 74.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a + \frac{b \cdot c}{d}}{\color{blue}{d}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} + -1 \cdot a}{d} \]

      3. associate-/l* N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} + -1 \cdot a}{d} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, \mathsf{neg}\left(a\right)\right)}{d} \]

      7. lower-neg.f64 73.6

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d} \]

   4. Applied rewrites 73.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} - a}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      5. lift-/.f64 73.6

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

   7. Applied rewrites 73.6%

      \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 71.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.11 \cdot 10^{+74}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.66 \cdot 10^{+75}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.11e+74)
   (/ b c)
   (if (<= c 1.66e+75) (/ (- (* c (/ b d)) a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.11e+74) {
		tmp = b / c;
	} else if (c <= 1.66e+75) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = b / c;
	}
	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(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.11d+74)) then
        tmp = b / c
    else if (c <= 1.66d+75) then
        tmp = ((c * (b / d)) - a) / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.11e+74) {
		tmp = b / c;
	} else if (c <= 1.66e+75) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.11e+74:
		tmp = b / c
	elif c <= 1.66e+75:
		tmp = ((c * (b / d)) - a) / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.11e+74)
		tmp = Float64(b / c);
	elseif (c <= 1.66e+75)
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.11e+74)
		tmp = b / c;
	elseif (c <= 1.66e+75)
		tmp = ((c * (b / d)) - a) / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.11e+74], N[(b / c), $MachinePrecision], If[LessEqual[c, 1.66e+75], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.11e74 or 1.66e75 < c

   1. Initial program 42.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b}{c}} \]
   3. Step-by-step derivation

      1. lower-/.f64 70.9

         \[\leadsto \frac{b}{\color{blue}{c}} \]

   4. Applied rewrites 70.9%

      \[\leadsto \color{blue}{\frac{b}{c}} \]

   if -1.11e74 < c < 1.66e75

   1. Initial program 74.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a + \frac{b \cdot c}{d}}{\color{blue}{d}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} + -1 \cdot a}{d} \]

      3. associate-/l* N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} + -1 \cdot a}{d} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, \mathsf{neg}\left(a\right)\right)}{d} \]

      7. lower-neg.f64 73.6

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d} \]

   4. Applied rewrites 73.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} - a}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      5. lift-/.f64 73.6

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

   7. Applied rewrites 73.6%

      \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{c}{d} \cdot b - a}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} - a}{d} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} - a}{d} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

      7. lift-*.f64 73.5

         \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

   9. Applied rewrites 73.5%

      \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\frac{c \cdot b}{d} - a}{d} \]

       3. associate-/l* N/A

          \[\leadsto \frac{c \cdot \frac{b}{d} - a}{d} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{c \cdot \frac{b}{d} - a}{d} \]

       5. lower-/.f64 72.0

          \[\leadsto \frac{c \cdot \frac{b}{d} - a}{d} \]

   11. Applied rewrites 72.0%

       \[\leadsto \frac{c \cdot \frac{b}{d} - a}{d} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 62.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.16 \cdot 10^{+73}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.16e+73) (/ b c) (if (<= c 2.2e+70) (/ (- a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.16e+73) {
		tmp = b / c;
	} else if (c <= 2.2e+70) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	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(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-2.16d+73)) then
        tmp = b / c
    else if (c <= 2.2d+70) then
        tmp = -a / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.16e+73) {
		tmp = b / c;
	} else if (c <= 2.2e+70) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -2.16e+73:
		tmp = b / c
	elif c <= 2.2e+70:
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.16e+73)
		tmp = Float64(b / c);
	elseif (c <= 2.2e+70)
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -2.16e+73)
		tmp = b / c;
	elseif (c <= 2.2e+70)
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.16e+73], N[(b / c), $MachinePrecision], If[LessEqual[c, 2.2e+70], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -2.1600000000000001e73 or 2.20000000000000001e70 < c

   1. Initial program 42.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b}{c}} \]
   3. Step-by-step derivation

      1. lower-/.f64 70.7

         \[\leadsto \frac{b}{\color{blue}{c}} \]

   4. Applied rewrites 70.7%

      \[\leadsto \color{blue}{\frac{b}{c}} \]

   if -2.1600000000000001e73 < c < 2.20000000000000001e70

   1. Initial program 74.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot a}{\color{blue}{d}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a}{\color{blue}{d}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(a\right)}{d} \]

      4. lower-neg.f64 57.1

         \[\leadsto \frac{-a}{d} \]

   4. Applied rewrites 57.1%

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

Alternative 9: 43.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \frac{b}{c} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (/ b c))

C

double code(double a, double b, double c, double d) {
	return b / c;
}

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(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = b / c
end function

Java

public static double code(double a, double b, double c, double d) {
	return b / c;
}

Python

def code(a, b, c, d):
	return b / c

Julia

function code(a, b, c, d)
	return Float64(b / c)
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = b / c;
end

Wolfram

code[a_, b_, c_, d_] := N[(b / c), $MachinePrecision]

Derivation

1. Initial program 62.4%

   \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

2. Taylor expanded in c around inf

   \[\leadsto \color{blue}{\frac{b}{c}} \]
3. Step-by-step derivation

   1. lower-/.f64 43.2

      \[\leadsto \frac{b}{\color{blue}{c}} \]

4. Applied rewrites 43.2%

   \[\leadsto \color{blue}{\frac{b}{c}} \]
5. Add Preprocessing

Developer Target 1: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (< (fabs d) (fabs c))
   (/ (- b (* a (/ d c))) (+ c (* d (/ d c))))
   (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (fabs(d) < fabs(c)) {
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (-a + (b * (c / d))) / (d + (c * (c / d)));
	}
	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(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (abs(d) < abs(c)) then
        tmp = (b - (a * (d / c))) / (c + (d * (d / c)))
    else
        tmp = (-a + (b * (c / d))) / (d + (c * (c / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (Math.abs(d) < Math.abs(c)) {
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (-a + (b * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if math.fabs(d) < math.fabs(c):
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (-a + (b * (c / d))) / (d + (c * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (abs(d) < abs(c))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(Float64(-a) + Float64(b * Float64(c / d))) / Float64(d + Float64(c * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (abs(d) < abs(c))
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)));
	else
		tmp = (-a + (b * (c / d))) / (d + (c * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Less[N[Abs[d], $MachinePrecision], N[Abs[c], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-a) + N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d + N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2025114 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (! :herbie-platform c (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d))))))

  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))