Complex division, imag part

Percentage Accurate: 61.6% → 82.6%

Time: 4.1s

Alternatives: 9

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: 61.6% 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: 82.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{if}\;c \leq -1.65 \cdot 10^{+129}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{-90}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 5.5 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 4 \cdot 10^{+96}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (fma c c (* d d)))))
   (if (<= c -1.65e+129)
     (/ (- b (* a (/ d c))) c)
     (if (<= c -5.5e+31)
       (/ (fma b (/ c d) (- a)) d)
       (if (<= c -1.1e-90)
         t_0
         (if (<= c 5.5e-152)
           (/ (- (/ (* b c) d) a) d)
           (if (<= c 4e+96) t_0 (/ (- b (* d (/ a c))) c))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / fma(c, c, (d * d));
	double tmp;
	if (c <= -1.65e+129) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= -5.5e+31) {
		tmp = fma(b, (c / d), -a) / d;
	} else if (c <= -1.1e-90) {
		tmp = t_0;
	} else if (c <= 5.5e-152) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 4e+96) {
		tmp = t_0;
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / fma(c, c, Float64(d * d)))
	tmp = 0.0
	if (c <= -1.65e+129)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= -5.5e+31)
		tmp = Float64(fma(b, Float64(c / d), Float64(-a)) / d);
	elseif (c <= -1.1e-90)
		tmp = t_0;
	elseif (c <= 5.5e-152)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 4e+96)
		tmp = t_0;
	else
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.65e+129], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -5.5e+31], N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, -1.1e-90], t$95$0, If[LessEqual[c, 5.5e-152], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 4e+96], t$95$0, N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -1.64999999999999995e129

   1. Initial program 34.9%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 80.3

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

   4. Applied rewrites 80.3%

      \[\leadsto \color{blue}{\frac{b - \frac{d \cdot a}{c}}{c}} \]
   5. 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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 86.0

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

   6. Applied rewrites 86.0%

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

   if -1.64999999999999995e129 < c < -5.50000000000000002e31

   1. Initial program 72.2%

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

   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 35.4

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

   4. Applied rewrites 35.4%

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

   if -5.50000000000000002e31 < c < -1.09999999999999993e-90 or 5.4999999999999998e-152 < c < 4.0000000000000002e96

   1. Initial program 76.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{c \cdot c + \color{blue}{{d}^{2}}} \]

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 76.8

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

   3. Applied rewrites 76.8%

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

   if -1.09999999999999993e-90 < c < 5.4999999999999998e-152

   1. Initial program 70.7%

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

   2. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{a + -1 \cdot \frac{b \cdot c}{d}}{d}} \]
   3. 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. lower-+.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 91.3

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

   4. Applied rewrites 91.3%

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

   5. Taylor expanded in d around inf

      \[\leadsto \frac{\frac{b \cdot c}{d} - a}{\color{blue}{d}} \]
   6. 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. lower-/.f64 N/A

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

      4. lower-*.f64 91.3

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

   7. Applied rewrites 91.3%

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

   if 4.0000000000000002e96 < c

   1. Initial program 35.7%

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

   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 78.2

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

   4. Applied rewrites 78.2%

      \[\leadsto \color{blue}{\frac{b - \frac{d \cdot a}{c}}{c}} \]
   5. 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 85.2

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

   6. Applied rewrites 85.2%

      \[\leadsto \frac{b - d \cdot \frac{a}{c}}{c} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 2: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.85 \cdot 10^{+142}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -1.25 \cdot 10^{-90}:\\ \;\;\;\;\left(-b\right) \cdot \frac{\mathsf{fma}\left(a, \frac{d}{b}, -c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 5.5 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 4 \cdot 10^{+96}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.85e+142)
   (/ (- b (* a (/ d c))) c)
   (if (<= c -1.25e-90)
     (* (- b) (/ (fma a (/ d b) (- c)) (fma d d (* c c))))
     (if (<= c 5.5e-152)
       (/ (- (/ (* b c) d) a) d)
       (if (<= c 4e+96)
         (/ (- (* b c) (* a d)) (fma c c (* d d)))
         (/ (- b (* d (/ a c))) c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.85e+142) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= -1.25e-90) {
		tmp = -b * (fma(a, (d / b), -c) / fma(d, d, (c * c)));
	} else if (c <= 5.5e-152) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 4e+96) {
		tmp = ((b * c) - (a * d)) / fma(c, c, (d * d));
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.85e+142)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= -1.25e-90)
		tmp = Float64(Float64(-b) * Float64(fma(a, Float64(d / b), Float64(-c)) / fma(d, d, Float64(c * c))));
	elseif (c <= 5.5e-152)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 4e+96)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / fma(c, c, Float64(d * 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, -1.85e+142], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -1.25e-90], N[((-b) * N[(N[(a * N[(d / b), $MachinePrecision] + (-c)), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 5.5e-152], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 4e+96], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -1.8499999999999999e142

   1. Initial program 33.1%

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

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   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 81.9

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

   4. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\frac{b - \frac{d \cdot a}{c}}{c}} \]
   5. 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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 88.1

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

   6. Applied rewrites 88.1%

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

   if -1.8499999999999999e142 < c < -1.25000000000000005e-90

   1. Initial program 73.8%

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

   2. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. associate-/r* N/A

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

      8. div-add-rev N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 68.4%

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

   if -1.25000000000000005e-90 < c < 5.4999999999999998e-152

   1. Initial program 70.7%

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

   2. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{a + -1 \cdot \frac{b \cdot c}{d}}{d}} \]
   3. 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. lower-+.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 91.3

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

   4. Applied rewrites 91.3%

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

   5. Taylor expanded in d around inf

      \[\leadsto \frac{\frac{b \cdot c}{d} - a}{\color{blue}{d}} \]
   6. 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. lower-/.f64 N/A

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

      4. lower-*.f64 91.3

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

   7. Applied rewrites 91.3%

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

   if 5.4999999999999998e-152 < c < 4.0000000000000002e96

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{c \cdot c + \color{blue}{{d}^{2}}} \]

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 76.9

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

   3. Applied rewrites 76.9%

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

   if 4.0000000000000002e96 < c

   1. Initial program 35.7%

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

   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 78.2

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

   4. Applied rewrites 78.2%

      \[\leadsto \color{blue}{\frac{b - \frac{d \cdot a}{c}}{c}} \]
   5. 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 85.2

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

   6. Applied rewrites 85.2%

      \[\leadsto \frac{b - d \cdot \frac{a}{c}}{c} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.65 \cdot 10^{+129}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;c \leq 7.8 \cdot 10^{+99}:\\ \;\;\;\;b \cdot \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.65e+129)
   (/ (- b (* a (/ d c))) c)
   (if (<= c 3e-6)
     (/ (fma b (/ c d) (- a)) d)
     (if (<= c 7.8e+99)
       (* b (/ c (fma d d (* c c))))
       (/ (- b (* d (/ a c))) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.65e+129) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 3e-6) {
		tmp = fma(b, (c / d), -a) / d;
	} else if (c <= 7.8e+99) {
		tmp = b * (c / fma(d, d, (c * c)));
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.65e+129)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= 3e-6)
		tmp = Float64(fma(b, Float64(c / d), Float64(-a)) / d);
	elseif (c <= 7.8e+99)
		tmp = Float64(b * Float64(c / fma(d, d, Float64(c * c))));
	else
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.65e+129], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 3e-6], N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 7.8e+99], N[(b * N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.64999999999999995e129

   1. Initial program 34.9%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 80.3

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

   4. Applied rewrites 80.3%

      \[\leadsto \color{blue}{\frac{b - \frac{d \cdot a}{c}}{c}} \]
   5. 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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 86.0

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

   6. Applied rewrites 86.0%

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

   if -1.64999999999999995e129 < c < 3.0000000000000001e-6

   1. Initial program 73.4%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 73.9

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

   4. Applied rewrites 73.9%

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

   if 3.0000000000000001e-6 < c < 7.79999999999999989e99

   1. Initial program 74.1%

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

   2. Taylor expanded in a around 0

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

      1. associate-/l* N/A

         \[\leadsto b \cdot \color{blue}{\frac{c}{{c}^{2} + {d}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto b \cdot \color{blue}{\frac{c}{{c}^{2} + {d}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto b \cdot \frac{c}{\color{blue}{{c}^{2} + {d}^{2}}} \]

      4. +-commutative N/A

         \[\leadsto b \cdot \frac{c}{{d}^{2} + \color{blue}{{c}^{2}}} \]

      5. pow2 N/A

         \[\leadsto b \cdot \frac{c}{d \cdot d + {\color{blue}{c}}^{2}} \]

      6. lower-fma.f64 N/A

         \[\leadsto b \cdot \frac{c}{\mathsf{fma}\left(d, \color{blue}{d}, {c}^{2}\right)} \]

      7. pow2 N/A

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

      8. lift-*.f64 60.8

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

   4. Applied rewrites 60.8%

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

   if 7.79999999999999989e99 < c

   1. Initial program 35.5%

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

   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 78.3

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

   4. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\frac{b - \frac{d \cdot a}{c}}{c}} \]
   5. 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 85.5

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

   6. Applied rewrites 85.5%

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

Alternative 4: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.1 \cdot 10^{+105}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 7.8 \cdot 10^{+99}:\\ \;\;\;\;b \cdot \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.1e+105)
   (/ (- b (* a (/ d c))) c)
   (if (<= c 3e-6)
     (/ (- (/ (* b c) d) a) d)
     (if (<= c 7.8e+99)
       (* b (/ c (fma d d (* c c))))
       (/ (- b (* d (/ a c))) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.1e+105) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 3e-6) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 7.8e+99) {
		tmp = b * (c / fma(d, d, (c * c)));
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.1e+105)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= 3e-6)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 7.8e+99)
		tmp = Float64(b * Float64(c / fma(d, d, Float64(c * c))));
	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+105], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 3e-6], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 7.8e+99], N[(b * N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.1000000000000001e105

   1. Initial program 38.0%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 80.1

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

   4. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\frac{b - \frac{d \cdot a}{c}}{c}} \]
   5. 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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 85.1

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

   6. Applied rewrites 85.1%

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

   if -2.1000000000000001e105 < c < 3.0000000000000001e-6

   1. Initial program 73.7%

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

   2. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{a + -1 \cdot \frac{b \cdot c}{d}}{d}} \]
   3. 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. lower-+.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 75.2

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

   4. Applied rewrites 75.2%

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

   5. Taylor expanded in d around inf

      \[\leadsto \frac{\frac{b \cdot c}{d} - a}{\color{blue}{d}} \]
   6. 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. lower-/.f64 N/A

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

      4. lower-*.f64 75.2

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

   7. Applied rewrites 75.2%

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

   if 3.0000000000000001e-6 < c < 7.79999999999999989e99

   1. Initial program 74.1%

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

   2. Taylor expanded in a around 0

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

      1. associate-/l* N/A

         \[\leadsto b \cdot \color{blue}{\frac{c}{{c}^{2} + {d}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto b \cdot \color{blue}{\frac{c}{{c}^{2} + {d}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto b \cdot \frac{c}{\color{blue}{{c}^{2} + {d}^{2}}} \]

      4. +-commutative N/A

         \[\leadsto b \cdot \frac{c}{{d}^{2} + \color{blue}{{c}^{2}}} \]

      5. pow2 N/A

         \[\leadsto b \cdot \frac{c}{d \cdot d + {\color{blue}{c}}^{2}} \]

      6. lower-fma.f64 N/A

         \[\leadsto b \cdot \frac{c}{\mathsf{fma}\left(d, \color{blue}{d}, {c}^{2}\right)} \]

      7. pow2 N/A

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

      8. lift-*.f64 60.8

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

   4. Applied rewrites 60.8%

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

   if 7.79999999999999989e99 < c

   1. Initial program 35.5%

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

   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 78.3

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

   4. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\frac{b - \frac{d \cdot a}{c}}{c}} \]
   5. 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 85.5

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

   6. Applied rewrites 85.5%

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

Alternative 5: 76.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{if}\;c \leq -2.1 \cdot 10^{+105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 7.8 \cdot 10^{+99}:\\ \;\;\;\;b \cdot \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* a (/ d c))) c)))
   (if (<= c -2.1e+105)
     t_0
     (if (<= c 3e-6)
       (/ (- (/ (* b c) d) a) d)
       (if (<= c 7.8e+99) (* b (/ c (fma d d (* c c)))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - (a * (d / c))) / c;
	double tmp;
	if (c <= -2.1e+105) {
		tmp = t_0;
	} else if (c <= 3e-6) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 7.8e+99) {
		tmp = b * (c / fma(d, d, (c * c)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(a * Float64(d / c))) / c)
	tmp = 0.0
	if (c <= -2.1e+105)
		tmp = t_0;
	elseif (c <= 3e-6)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 7.8e+99)
		tmp = Float64(b * Float64(c / fma(d, d, Float64(c * c))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -2.1e+105], t$95$0, If[LessEqual[c, 3e-6], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 7.8e+99], N[(b * N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.1000000000000001e105 or 7.79999999999999989e99 < c

   1. Initial program 36.7%

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

   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 79.2

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

   4. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\frac{b - \frac{d \cdot a}{c}}{c}} \]
   5. 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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 84.6

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

   6. Applied rewrites 84.6%

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

   if -2.1000000000000001e105 < c < 3.0000000000000001e-6

   1. Initial program 73.7%

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

   2. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{a + -1 \cdot \frac{b \cdot c}{d}}{d}} \]
   3. 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. lower-+.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 75.2

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

   4. Applied rewrites 75.2%

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

   5. Taylor expanded in d around inf

      \[\leadsto \frac{\frac{b \cdot c}{d} - a}{\color{blue}{d}} \]
   6. 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. lower-/.f64 N/A

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

      4. lower-*.f64 75.2

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

   7. Applied rewrites 75.2%

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

   if 3.0000000000000001e-6 < c < 7.79999999999999989e99

   1. Initial program 74.1%

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

   2. Taylor expanded in a around 0

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

      1. associate-/l* N/A

         \[\leadsto b \cdot \color{blue}{\frac{c}{{c}^{2} + {d}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto b \cdot \color{blue}{\frac{c}{{c}^{2} + {d}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto b \cdot \frac{c}{\color{blue}{{c}^{2} + {d}^{2}}} \]

      4. +-commutative N/A

         \[\leadsto b \cdot \frac{c}{{d}^{2} + \color{blue}{{c}^{2}}} \]

      5. pow2 N/A

         \[\leadsto b \cdot \frac{c}{d \cdot d + {\color{blue}{c}}^{2}} \]

      6. lower-fma.f64 N/A

         \[\leadsto b \cdot \frac{c}{\mathsf{fma}\left(d, \color{blue}{d}, {c}^{2}\right)} \]

      7. pow2 N/A

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

      8. lift-*.f64 60.8

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

   4. Applied rewrites 60.8%

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

Alternative 6: 74.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7 \cdot 10^{+97}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+140}:\\ \;\;\;\;b \cdot \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -7e+97)
   (/ b c)
   (if (<= c 3e-6)
     (/ (- (/ (* b c) d) a) d)
     (if (<= c 2.7e+140) (* b (/ c (fma d d (* c c)))) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7e+97) {
		tmp = b / c;
	} else if (c <= 3e-6) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 2.7e+140) {
		tmp = b * (c / fma(d, d, (c * c)));
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -7e+97)
		tmp = Float64(b / c);
	elseif (c <= 3e-6)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 2.7e+140)
		tmp = Float64(b * Float64(c / fma(d, d, Float64(c * c))));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -7e+97], N[(b / c), $MachinePrecision], If[LessEqual[c, 3e-6], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.7e+140], N[(b * N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.0000000000000001e97 or 2.70000000000000018e140 < c

   1. Initial program 34.6%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 75.2

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

   4. Applied rewrites 75.2%

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

   if -7.0000000000000001e97 < c < 3.0000000000000001e-6

   1. Initial program 73.8%

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

   2. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{a + -1 \cdot \frac{b \cdot c}{d}}{d}} \]
   3. 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. lower-+.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 75.8

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

   4. Applied rewrites 75.8%

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

   5. Taylor expanded in d around inf

      \[\leadsto \frac{\frac{b \cdot c}{d} - a}{\color{blue}{d}} \]
   6. 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. lower-/.f64 N/A

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

      4. lower-*.f64 75.8

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

   7. Applied rewrites 75.8%

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

   if 3.0000000000000001e-6 < c < 2.70000000000000018e140

   1. Initial program 72.2%

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

   2. Taylor expanded in a around 0

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

      1. associate-/l* N/A

         \[\leadsto b \cdot \color{blue}{\frac{c}{{c}^{2} + {d}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto b \cdot \color{blue}{\frac{c}{{c}^{2} + {d}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto b \cdot \frac{c}{\color{blue}{{c}^{2} + {d}^{2}}} \]

      4. +-commutative N/A

         \[\leadsto b \cdot \frac{c}{{d}^{2} + \color{blue}{{c}^{2}}} \]

      5. pow2 N/A

         \[\leadsto b \cdot \frac{c}{d \cdot d + {\color{blue}{c}}^{2}} \]

      6. lower-fma.f64 N/A

         \[\leadsto b \cdot \frac{c}{\mathsf{fma}\left(d, \color{blue}{d}, {c}^{2}\right)} \]

      7. pow2 N/A

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

      8. lift-*.f64 62.4

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

   4. Applied rewrites 62.4%

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

Alternative 7: 65.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6.6 \cdot 10^{+61}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+140}:\\ \;\;\;\;b \cdot \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -6.6e+61)
   (/ b c)
   (if (<= c 1.2e-125)
     (/ (- a) d)
     (if (<= c 2.7e+140) (* b (/ c (fma d d (* c c)))) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -6.6e+61) {
		tmp = b / c;
	} else if (c <= 1.2e-125) {
		tmp = -a / d;
	} else if (c <= 2.7e+140) {
		tmp = b * (c / fma(d, d, (c * c)));
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -6.6e+61)
		tmp = Float64(b / c);
	elseif (c <= 1.2e-125)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 2.7e+140)
		tmp = Float64(b * Float64(c / fma(d, d, Float64(c * c))));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -6.6e+61], N[(b / c), $MachinePrecision], If[LessEqual[c, 1.2e-125], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 2.7e+140], N[(b * N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -6.5999999999999995e61 or 2.70000000000000018e140 < c

   1. Initial program 38.2%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 72.9

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

   4. Applied rewrites 72.9%

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

   if -6.5999999999999995e61 < c < 1.2000000000000001e-125

   1. Initial program 72.7%

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

   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 64.6

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

   4. Applied rewrites 64.6%

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

   if 1.2000000000000001e-125 < c < 2.70000000000000018e140

   1. Initial program 75.2%

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

   2. Taylor expanded in a around 0

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

      1. associate-/l* N/A

         \[\leadsto b \cdot \color{blue}{\frac{c}{{c}^{2} + {d}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto b \cdot \color{blue}{\frac{c}{{c}^{2} + {d}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto b \cdot \frac{c}{\color{blue}{{c}^{2} + {d}^{2}}} \]

      4. +-commutative N/A

         \[\leadsto b \cdot \frac{c}{{d}^{2} + \color{blue}{{c}^{2}}} \]

      5. pow2 N/A

         \[\leadsto b \cdot \frac{c}{d \cdot d + {\color{blue}{c}}^{2}} \]

      6. lower-fma.f64 N/A

         \[\leadsto b \cdot \frac{c}{\mathsf{fma}\left(d, \color{blue}{d}, {c}^{2}\right)} \]

      7. pow2 N/A

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

      8. lift-*.f64 56.0

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

   4. Applied rewrites 56.0%

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

Alternative 8: 64.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6.6 \cdot 10^{+61}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -6.6e+61) (/ b c) (if (<= c 4.5e-33) (/ (- a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -6.6e+61) {
		tmp = b / c;
	} else if (c <= 4.5e-33) {
		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 <= (-6.6d+61)) then
        tmp = b / c
    else if (c <= 4.5d-33) 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 <= -6.6e+61) {
		tmp = b / c;
	} else if (c <= 4.5e-33) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -6.6e+61:
		tmp = b / c
	elif c <= 4.5e-33:
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -6.6e+61)
		tmp = Float64(b / c);
	elseif (c <= 4.5e-33)
		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 <= -6.6e+61)
		tmp = b / c;
	elseif (c <= 4.5e-33)
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -6.6e+61], N[(b / c), $MachinePrecision], If[LessEqual[c, 4.5e-33], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -6.5999999999999995e61 or 4.49999999999999991e-33 < c

   1. Initial program 48.2%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 66.0

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

   4. Applied rewrites 66.0%

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

   if -6.5999999999999995e61 < c < 4.49999999999999991e-33

   1. Initial program 73.7%

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

   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 62.5

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

   4. Applied rewrites 62.5%

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

Alternative 9: 43.4% 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 61.6%

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

2. Taylor expanded in c around inf

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

   1. lower-/.f64 43.4

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

4. Applied rewrites 43.4%

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

Reproduce

herbie shell --seed 2025101 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64
  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))