Complex division, imag part

Percentage Accurate: 63.4% → 80.8%

Time: 3.6s

Alternatives: 7

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: 63.4% 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: 80.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.6 \cdot 10^{+96}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{c}\\ \mathbf{elif}\;c \leq -3.9 \cdot 10^{-122}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 10^{-39}:\\ \;\;\;\;\frac{\frac{c}{d} \cdot b - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \frac{b}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -7.6e+96)
   (- (/ (fma a (/ d c) (- b)) c))
   (if (<= c -3.9e-122)
     (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))
     (if (<= c 1e-39)
       (/ (- (* (/ c d) b) a) d)
       (fma (/ (- a) c) (/ d c) (/ b c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7.6e+96) {
		tmp = -(fma(a, (d / c), -b) / c);
	} else if (c <= -3.9e-122) {
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
	} else if (c <= 1e-39) {
		tmp = (((c / d) * b) - a) / d;
	} else {
		tmp = fma((-a / c), (d / c), (b / c));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -7.6e+96)
		tmp = Float64(-Float64(fma(a, Float64(d / c), Float64(-b)) / c));
	elseif (c <= -3.9e-122)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 1e-39)
		tmp = Float64(Float64(Float64(Float64(c / d) * b) - a) / d);
	else
		tmp = fma(Float64(Float64(-a) / c), Float64(d / c), Float64(b / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -7.6e+96], (-N[(N[(a * N[(d / c), $MachinePrecision] + (-b)), $MachinePrecision] / c), $MachinePrecision]), If[LessEqual[c, -3.9e-122], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1e-39], N[(N[(N[(N[(c / d), $MachinePrecision] * b), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[((-a) / c), $MachinePrecision] * N[(d / c), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -7.6000000000000003e96

   1. Initial program 42.5%

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

   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 83.5

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

   4. Applied rewrites 83.5%

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

   if -7.6000000000000003e96 < c < -3.8999999999999999e-122

   1. Initial program 77.8%

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

   if -3.8999999999999999e-122 < c < 9.99999999999999929e-40

   1. Initial program 72.5%

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

   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 87.1

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

   4. Applied rewrites 87.1%

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

   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 87.1

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

   7. Applied rewrites 87.1%

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

   if 9.99999999999999929e-40 < c

   1. Initial program 54.8%

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

   2. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 72.7

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

   4. Applied rewrites 72.7%

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

Alternative 2: 78.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \frac{b}{c}\right)\\ \mathbf{if}\;c \leq -430000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 10^{-39}:\\ \;\;\;\;\frac{\frac{c}{d} \cdot b - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma (/ (- a) c) (/ d c) (/ b c))))
   (if (<= c -430000000.0)
     t_0
     (if (<= c 1e-39) (/ (- (* (/ c d) b) a) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((-a / c), (d / c), (b / c));
	double tmp;
	if (c <= -430000000.0) {
		tmp = t_0;
	} else if (c <= 1e-39) {
		tmp = (((c / d) * b) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(Float64(Float64(-a) / c), Float64(d / c), Float64(b / c))
	tmp = 0.0
	if (c <= -430000000.0)
		tmp = t_0;
	elseif (c <= 1e-39)
		tmp = Float64(Float64(Float64(Float64(c / d) * b) - a) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[((-a) / c), $MachinePrecision] * N[(d / c), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -430000000.0], t$95$0, If[LessEqual[c, 1e-39], N[(N[(N[(N[(c / d), $MachinePrecision] * b), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -4.3e8 or 9.99999999999999929e-40 < c

   1. Initial program 53.3%

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

   2. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 74.3

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

   4. Applied rewrites 74.3%

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

   if -4.3e8 < c < 9.99999999999999929e-40

   1. Initial program 74.3%

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

   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 81.4

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

   4. Applied rewrites 81.4%

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

   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 81.4

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

   7. Applied rewrites 81.4%

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

Alternative 3: 77.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{c}\\ \mathbf{if}\;c \leq -430000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 6.4 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{c}{d} \cdot b - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (/ (fma a (/ d c) (- b)) c))))
   (if (<= c -430000000.0)
     t_0
     (if (<= c 6.4e-40) (/ (- (* (/ c d) b) a) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -(fma(a, (d / c), -b) / c);
	double tmp;
	if (c <= -430000000.0) {
		tmp = t_0;
	} else if (c <= 6.4e-40) {
		tmp = (((c / d) * b) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(-Float64(fma(a, Float64(d / c), Float64(-b)) / c))
	tmp = 0.0
	if (c <= -430000000.0)
		tmp = t_0;
	elseif (c <= 6.4e-40)
		tmp = Float64(Float64(Float64(Float64(c / d) * b) - a) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = (-N[(N[(a * N[(d / c), $MachinePrecision] + (-b)), $MachinePrecision] / c), $MachinePrecision])}, If[LessEqual[c, -430000000.0], t$95$0, If[LessEqual[c, 6.4e-40], N[(N[(N[(N[(c / d), $MachinePrecision] * b), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -4.3e8 or 6.40000000000000004e-40 < c

   1. Initial program 53.3%

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

   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 74.0

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

   4. Applied rewrites 74.0%

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

   if -4.3e8 < c < 6.40000000000000004e-40

   1. Initial program 74.3%

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

   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 81.4

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

   4. Applied rewrites 81.4%

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

   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 81.4

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

   7. Applied rewrites 81.4%

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

Alternative 4: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{c}{d} \cdot b - a}{d}\\ \mathbf{if}\;d \leq -1.95 \cdot 10^{+48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-11}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* (/ c d) b) a) d)))
   (if (<= d -1.95e+48) t_0 (if (<= d 7e-11) (/ (- b (/ (* d a) c)) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (((c / d) * b) - a) / d;
	double tmp;
	if (d <= -1.95e+48) {
		tmp = t_0;
	} else if (d <= 7e-11) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = t_0;
	}
	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) :: tmp
    t_0 = (((c / d) * b) - a) / d
    if (d <= (-1.95d+48)) then
        tmp = t_0
    else if (d <= 7d-11) then
        tmp = (b - ((d * a) / c)) / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (((c / d) * b) - a) / d;
	double tmp;
	if (d <= -1.95e+48) {
		tmp = t_0;
	} else if (d <= 7e-11) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (((c / d) * b) - a) / d
	tmp = 0
	if d <= -1.95e+48:
		tmp = t_0
	elif d <= 7e-11:
		tmp = (b - ((d * a) / c)) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(Float64(c / d) * b) - a) / d)
	tmp = 0.0
	if (d <= -1.95e+48)
		tmp = t_0;
	elseif (d <= 7e-11)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (((c / d) * b) - a) / d;
	tmp = 0.0;
	if (d <= -1.95e+48)
		tmp = t_0;
	elseif (d <= 7e-11)
		tmp = (b - ((d * a) / c)) / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(N[(c / d), $MachinePrecision] * b), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.95e+48], t$95$0, If[LessEqual[d, 7e-11], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.95e48 or 7.00000000000000038e-11 < d

   1. Initial program 50.6%

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

   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 78.1

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

   4. Applied rewrites 78.1%

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

   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 78.1

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

   7. Applied rewrites 78.1%

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

   if -1.95e48 < d < 7.00000000000000038e-11

   1. Initial program 74.8%

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

   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 79.1

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

   4. Applied rewrites 79.1%

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

Alternative 5: 72.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.1 \cdot 10^{+97}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 15.2:\\ \;\;\;\;\frac{\frac{c}{d} \cdot b - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.1e+97)
   (/ b c)
   (if (<= c 15.2) (/ (- (* (/ c d) b) a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.1e+97) {
		tmp = b / c;
	} else if (c <= 15.2) {
		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 <= (-2.1d+97)) then
        tmp = b / c
    else if (c <= 15.2d0) 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 <= -2.1e+97) {
		tmp = b / c;
	} else if (c <= 15.2) {
		tmp = (((c / d) * b) - a) / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -2.1e+97:
		tmp = b / c
	elif c <= 15.2:
		tmp = (((c / d) * b) - a) / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.1e+97)
		tmp = Float64(b / c);
	elseif (c <= 15.2)
		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 <= -2.1e+97)
		tmp = b / c;
	elseif (c <= 15.2)
		tmp = (((c / d) * b) - a) / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.1e+97], N[(b / c), $MachinePrecision], If[LessEqual[c, 15.2], 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 < -2.10000000000000012e97 or 15.199999999999999 < c

   1. Initial program 47.9%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 67.8

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

   4. Applied rewrites 67.8%

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

   if -2.10000000000000012e97 < c < 15.199999999999999

   1. Initial program 74.5%

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

   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 75.8

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

   4. Applied rewrites 75.8%

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

   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 75.8

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

   7. Applied rewrites 75.8%

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

Alternative 6: 62.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.35 \cdot 10^{+96}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{-38}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.35e+96) (/ b c) (if (<= c 5.2e-38) (/ (- a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.35e+96) {
		tmp = b / c;
	} else if (c <= 5.2e-38) {
		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 <= (-1.35d+96)) then
        tmp = b / c
    else if (c <= 5.2d-38) 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 <= -1.35e+96) {
		tmp = b / c;
	} else if (c <= 5.2e-38) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.35e+96:
		tmp = b / c
	elif c <= 5.2e-38:
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.35e+96)
		tmp = Float64(b / c);
	elseif (c <= 5.2e-38)
		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 <= -1.35e+96)
		tmp = b / c;
	elseif (c <= 5.2e-38)
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.35e+96], N[(b / c), $MachinePrecision], If[LessEqual[c, 5.2e-38], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.35000000000000011e96 or 5.20000000000000022e-38 < c

   1. Initial program 50.1%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 65.4

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

   4. Applied rewrites 65.4%

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

   if -1.35000000000000011e96 < c < 5.20000000000000022e-38

   1. Initial program 74.2%

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

   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 60.0

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

   4. Applied rewrites 60.0%

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

Alternative 7: 42.5% 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 63.4%

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

2. Taylor expanded in c around inf

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

   1. lower-/.f64 42.5

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

4. Applied rewrites 42.5%

   \[\leadsto \color{blue}{\frac{b}{c}} \]
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 2025093 
(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))))