Complex division, imag part

Percentage Accurate: 61.3% → 82.1%

Time: 5.6s

Alternatives: 7

Speedup: 1.5×

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: 61.3% 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: 82.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{if}\;c \leq -6 \cdot 10^{+52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -3.8 \cdot 10^{-137}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a\\ \mathbf{elif}\;c \leq 6 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 3.65 \cdot 10^{+59}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* d (/ a c))) c)))
   (if (<= c -6e+52)
     t_0
     (if (<= c -3.8e-137)
       (* (/ (fma b (/ c a) (- d)) (fma d d (* c c))) a)
       (if (<= c 6e-161)
         (/ (- (/ (* c b) d) a) d)
         (if (<= c 3.65e+59)
           (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))
           t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - (d * (a / c))) / c;
	double tmp;
	if (c <= -6e+52) {
		tmp = t_0;
	} else if (c <= -3.8e-137) {
		tmp = (fma(b, (c / a), -d) / fma(d, d, (c * c))) * a;
	} else if (c <= 6e-161) {
		tmp = (((c * b) / d) - a) / d;
	} else if (c <= 3.65e+59) {
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(d * Float64(a / c))) / c)
	tmp = 0.0
	if (c <= -6e+52)
		tmp = t_0;
	elseif (c <= -3.8e-137)
		tmp = Float64(Float64(fma(b, Float64(c / a), Float64(-d)) / fma(d, d, Float64(c * c))) * a);
	elseif (c <= 6e-161)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	elseif (c <= 3.65e+59)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -6e+52], t$95$0, If[LessEqual[c, -3.8e-137], N[(N[(N[(b * N[(c / a), $MachinePrecision] + (-d)), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[c, 6e-161], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 3.65e+59], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -6e52 or 3.6500000000000001e59 < c

   1. Initial program 37.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. 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 80.3

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

   5. Applied rewrites 80.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 85.8

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

   7. Applied rewrites 85.8%

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

   if -6e52 < c < -3.79999999999999999e-137

   1. Initial program 85.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \frac{d}{{c}^{2} + {d}^{2}} + \frac{b \cdot c}{a \cdot \left({c}^{2} + {d}^{2}\right)}\right)} \]
   4. 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

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

   5. Applied rewrites 88.3%

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

   if -3.79999999999999999e-137 < c < 5.99999999999999977e-161

   1. Initial program 69.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 96.0

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 96.0

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

   8. Applied rewrites 96.0%

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

   if 5.99999999999999977e-161 < c < 3.6500000000000001e59

   1. Initial program 83.1%

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

Alternative 2: 83.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{if}\;c \leq -6 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -2.9 \cdot 10^{-137}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 6 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 3.65 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))
        (t_1 (/ (- b (* d (/ a c))) c)))
   (if (<= c -6e+52)
     t_1
     (if (<= c -2.9e-137)
       t_0
       (if (<= c 6e-161)
         (/ (- (/ (* c b) d) a) d)
         (if (<= c 3.65e+59) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = (b - (d * (a / c))) / c;
	double tmp;
	if (c <= -6e+52) {
		tmp = t_1;
	} else if (c <= -2.9e-137) {
		tmp = t_0;
	} else if (c <= 6e-161) {
		tmp = (((c * b) / d) - a) / d;
	} else if (c <= 3.65e+59) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
    t_1 = (b - (d * (a / c))) / c
    if (c <= (-6d+52)) then
        tmp = t_1
    else if (c <= (-2.9d-137)) then
        tmp = t_0
    else if (c <= 6d-161) then
        tmp = (((c * b) / d) - a) / d
    else if (c <= 3.65d+59) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = (b - (d * (a / c))) / c;
	double tmp;
	if (c <= -6e+52) {
		tmp = t_1;
	} else if (c <= -2.9e-137) {
		tmp = t_0;
	} else if (c <= 6e-161) {
		tmp = (((c * b) / d) - a) / d;
	} else if (c <= 3.65e+59) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
	t_1 = (b - (d * (a / c))) / c
	tmp = 0
	if c <= -6e+52:
		tmp = t_1
	elif c <= -2.9e-137:
		tmp = t_0
	elif c <= 6e-161:
		tmp = (((c * b) / d) - a) / d
	elif c <= 3.65e+59:
		tmp = t_0
	else:
		tmp = t_1
	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)))
	t_1 = Float64(Float64(b - Float64(d * Float64(a / c))) / c)
	tmp = 0.0
	if (c <= -6e+52)
		tmp = t_1;
	elseif (c <= -2.9e-137)
		tmp = t_0;
	elseif (c <= 6e-161)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	elseif (c <= 3.65e+59)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	t_1 = (b - (d * (a / c))) / c;
	tmp = 0.0;
	if (c <= -6e+52)
		tmp = t_1;
	elseif (c <= -2.9e-137)
		tmp = t_0;
	elseif (c <= 6e-161)
		tmp = (((c * b) / d) - a) / d;
	elseif (c <= 3.65e+59)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = 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]}, Block[{t$95$1 = N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -6e+52], t$95$1, If[LessEqual[c, -2.9e-137], t$95$0, If[LessEqual[c, 6e-161], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 3.65e+59], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -6e52 or 3.6500000000000001e59 < c

   1. Initial program 37.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. 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 80.3

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

   5. Applied rewrites 80.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 85.8

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

   7. Applied rewrites 85.8%

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

   if -6e52 < c < -2.89999999999999985e-137 or 5.99999999999999977e-161 < c < 3.6500000000000001e59

   1. Initial program 84.2%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   if -2.89999999999999985e-137 < c < 5.99999999999999977e-161

   1. Initial program 69.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 96.0

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 96.0

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

   8. Applied rewrites 96.0%

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

Alternative 3: 71.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.45 \cdot 10^{+136}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -2.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c}\\ \mathbf{elif}\;c \leq 4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.45e+136)
   (/ b c)
   (if (<= c -2.2e-79)
     (/ (- (* b c) (* a d)) (* c c))
     (if (<= c 4e-13) (/ (- (/ (* c b) d) a) d) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.45e+136) {
		tmp = b / c;
	} else if (c <= -2.2e-79) {
		tmp = ((b * c) - (a * d)) / (c * c);
	} else if (c <= 4e-13) {
		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 <= (-2.45d+136)) then
        tmp = b / c
    else if (c <= (-2.2d-79)) then
        tmp = ((b * c) - (a * d)) / (c * c)
    else if (c <= 4d-13) 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 <= -2.45e+136) {
		tmp = b / c;
	} else if (c <= -2.2e-79) {
		tmp = ((b * c) - (a * d)) / (c * c);
	} else if (c <= 4e-13) {
		tmp = (((c * b) / d) - a) / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -2.45e+136:
		tmp = b / c
	elif c <= -2.2e-79:
		tmp = ((b * c) - (a * d)) / (c * c)
	elif c <= 4e-13:
		tmp = (((c * b) / d) - a) / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.45e+136)
		tmp = Float64(b / c);
	elseif (c <= -2.2e-79)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(c * c));
	elseif (c <= 4e-13)
		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 <= -2.45e+136)
		tmp = b / c;
	elseif (c <= -2.2e-79)
		tmp = ((b * c) - (a * d)) / (c * c);
	elseif (c <= 4e-13)
		tmp = (((c * b) / d) - a) / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.45e+136], N[(b / c), $MachinePrecision], If[LessEqual[c, -2.2e-79], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4e-13], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.45000000000000006e136 or 4.0000000000000001e-13 < c

   1. Initial program 43.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 70.9

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

   5. Applied rewrites 70.9%

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

   if -2.45000000000000006e136 < c < -2.1999999999999999e-79

   1. Initial program 75.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{c}^{2}}} \]
   4. Step-by-step derivation

      1. pow2 N/A

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

      2. lift-*.f64 66.7

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

   5. Applied rewrites 66.7%

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

   if -2.1999999999999999e-79 < c < 4.0000000000000001e-13

   1. Initial program 74.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 85.3

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

   8. Applied rewrites 85.3%

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

Alternative 4: 77.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.5 \cdot 10^{-76} \lor \neg \left(c \leq 6.5 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -3.5e-76) (not (<= c 6.5e-20)))
   (/ (- b (* d (/ a c))) c)
   (/ (- (/ (* c b) d) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.5e-76) || !(c <= 6.5e-20)) {
		tmp = (b - (d * (a / c))) / c;
	} else {
		tmp = (((c * b) / d) - 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 ((c <= (-3.5d-76)) .or. (.not. (c <= 6.5d-20))) then
        tmp = (b - (d * (a / c))) / c
    else
        tmp = (((c * b) / d) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.5e-76) || !(c <= 6.5e-20)) {
		tmp = (b - (d * (a / c))) / c;
	} else {
		tmp = (((c * b) / d) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -3.5e-76) or not (c <= 6.5e-20):
		tmp = (b - (d * (a / c))) / c
	else:
		tmp = (((c * b) / d) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -3.5e-76) || !(c <= 6.5e-20))
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	else
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -3.5e-76) || ~((c <= 6.5e-20)))
		tmp = (b - (d * (a / c))) / c;
	else
		tmp = (((c * b) / d) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -3.5e-76], N[Not[LessEqual[c, 6.5e-20]], $MachinePrecision]], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.49999999999999997e-76 or 6.50000000000000032e-20 < c

   1. Initial program 50.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. 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 78.2

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

   5. Applied rewrites 78.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 81.7

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

   7. Applied rewrites 81.7%

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

   if -3.49999999999999997e-76 < c < 6.50000000000000032e-20

   1. Initial program 74.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 85.3

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

   8. Applied rewrites 85.3%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.5 \cdot 10^{-76} \lor \neg \left(c \leq 6.5 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.1 \cdot 10^{-79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{-c}\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.1e-79)
   (/ (fma a (/ d c) (- b)) (- c))
   (if (<= c 6.5e-20) (/ (- (/ (* c b) d) a) d) (/ (- b (* d (/ a c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.1e-79) {
		tmp = fma(a, (d / c), -b) / -c;
	} else if (c <= 6.5e-20) {
		tmp = (((c * b) / d) - a) / d;
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.1e-79)
		tmp = Float64(fma(a, Float64(d / c), Float64(-b)) / Float64(-c));
	elseif (c <= 6.5e-20)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.1e-79], N[(N[(a * N[(d / c), $MachinePrecision] + (-b)), $MachinePrecision] / (-c)), $MachinePrecision], If[LessEqual[c, 6.5e-20], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.0999999999999999e-79

   1. Initial program 49.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot b + \frac{a \cdot d}{c}}{c}} \]
   4. 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 83.8

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

   5. Applied rewrites 83.8%

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

   if -2.0999999999999999e-79 < c < 6.50000000000000032e-20

   1. Initial program 73.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 86.1

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

   5. Applied rewrites 86.1%

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

   6. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 86.1

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

   8. Applied rewrites 86.1%

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

   if 6.50000000000000032e-20 < c

   1. Initial program 52.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. 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 77.7

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

   5. Applied rewrites 77.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 80.5

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

   7. Applied rewrites 80.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.1 \cdot 10^{-79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{-c}\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{+71} \lor \neg \left(d \leq 3.15 \cdot 10^{+50}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2e+71) (not (<= d 3.15e+50))) (/ (- a) d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2e+71) || !(d <= 3.15e+50)) {
		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 ((d <= (-2d+71)) .or. (.not. (d <= 3.15d+50))) 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 ((d <= -2e+71) || !(d <= 3.15e+50)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2e+71) or not (d <= 3.15e+50):
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2e+71) || !(d <= 3.15e+50))
		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 ((d <= -2e+71) || ~((d <= 3.15e+50)))
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2e+71], N[Not[LessEqual[d, 3.15e+50]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.0000000000000001e71 or 3.14999999999999993e50 < d

   1. Initial program 39.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   4. 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 68.0

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

   5. Applied rewrites 68.0%

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

   if -2.0000000000000001e71 < d < 3.14999999999999993e50

   1. Initial program 72.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 65.9

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

   5. Applied rewrites 65.9%

      \[\leadsto \color{blue}{\frac{b}{c}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{+71} \lor \neg \left(d \leq 3.15 \cdot 10^{+50}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.2% accurate, 3.2× 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 60.6%

   \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
2. Add Preprocessing

3. Taylor expanded in c around inf

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

   1. lower-/.f64 48.6

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

5. Applied rewrites 48.6%

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

Reproduce

herbie shell --seed 2025085 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (! :herbie-platform default (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))))