Complex division, imag part

Percentage Accurate: 62.1% → 81.1%

Time: 5.3s

Alternatives: 6

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: 62.1% 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: 81.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -2.7 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{-146}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.48 \cdot 10^{+51}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, d, b \cdot c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -2.7e-26)
     t_0
     (if (<= d 7.8e-146)
       (/ (- b (/ (* a d) c)) c)
       (if (<= d 1.48e+51) (/ (fma (- a) d (* b c)) (fma d d (* c c))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(c, (b / d), -a) / d;
	double tmp;
	if (d <= -2.7e-26) {
		tmp = t_0;
	} else if (d <= 7.8e-146) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 1.48e+51) {
		tmp = fma(-a, d, (b * c)) / fma(d, d, (c * c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(c, Float64(b / d), Float64(-a)) / d)
	tmp = 0.0
	if (d <= -2.7e-26)
		tmp = t_0;
	elseif (d <= 7.8e-146)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 1.48e+51)
		tmp = Float64(fma(Float64(-a), d, Float64(b * 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[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2.7e-26], t$95$0, If[LessEqual[d, 7.8e-146], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.48e+51], N[(N[((-a) * d + N[(b * c), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.69999999999999982e-26 or 1.48e51 < d

   1. Initial program 49.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 49.1

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqr-abs-rev N/A

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

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

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

      12. rem-sqrt-square-rev N/A

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

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

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 49.1%

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

   5. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. div-add N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 86.0

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

   7. Applied rewrites 86.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}} \]
   8. Step-by-step derivation

   9. Applied rewrites 86.0%

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

   if -2.69999999999999982e-26 < d < 7.80000000000000005e-146

   1. Initial program 66.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 90.6

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

   5. Applied rewrites 90.6%

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

   if 7.80000000000000005e-146 < d < 1.48e51

   1. Initial program 87.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 87.5

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqr-abs-rev N/A

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

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

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

      12. rem-sqrt-square-rev N/A

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

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

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 87.6%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.7 \cdot 10^{-26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{-146}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.48 \cdot 10^{+51}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, d, b \cdot c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -2.7 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{-46}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 8 \cdot 10^{+68}:\\ \;\;\;\;\frac{\left(-d\right) \cdot a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -2.7e-26)
     t_0
     (if (<= d 6.8e-46)
       (/ (- b (/ (* a d) c)) c)
       (if (<= d 8e+68) (/ (* (- d) a) (fma d d (* c c))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(c, (b / d), -a) / d;
	double tmp;
	if (d <= -2.7e-26) {
		tmp = t_0;
	} else if (d <= 6.8e-46) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 8e+68) {
		tmp = (-d * a) / fma(d, d, (c * c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(c, Float64(b / d), Float64(-a)) / d)
	tmp = 0.0
	if (d <= -2.7e-26)
		tmp = t_0;
	elseif (d <= 6.8e-46)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 8e+68)
		tmp = Float64(Float64(Float64(-d) * a) / 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[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2.7e-26], t$95$0, If[LessEqual[d, 6.8e-46], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 8e+68], N[(N[((-d) * a), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.69999999999999982e-26 or 7.99999999999999962e68 < d

   1. Initial program 48.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 48.7

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqr-abs-rev N/A

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

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

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

      12. rem-sqrt-square-rev N/A

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

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

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 48.7%

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

   5. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. div-add N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 86.4

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

   7. Applied rewrites 86.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}} \]
   8. Step-by-step derivation

   9. Applied rewrites 86.5%

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

   if -2.69999999999999982e-26 < d < 6.79999999999999992e-46

   1. Initial program 69.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 87.3

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

   5. Applied rewrites 87.3%

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

   if 6.79999999999999992e-46 < d < 7.99999999999999962e68

   1. Initial program 84.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 84.2

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqr-abs-rev N/A

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

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

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

      12. rem-sqrt-square-rev N/A

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

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

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 84.2%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 73.2

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

   7. Applied rewrites 73.2%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.7 \cdot 10^{-26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{-46}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 8 \cdot 10^{+68}:\\ \;\;\;\;\frac{\left(-d\right) \cdot a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -4.8 \cdot 10^{-27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{+69}:\\ \;\;\;\;\left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -4.8e-27)
     t_0
     (if (<= d 8.5e-93)
       (/ b c)
       (if (<= d 1.05e+69) (* (- a) (/ d (fma d d (* c c)))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(c, (b / d), -a) / d;
	double tmp;
	if (d <= -4.8e-27) {
		tmp = t_0;
	} else if (d <= 8.5e-93) {
		tmp = b / c;
	} else if (d <= 1.05e+69) {
		tmp = -a * (d / fma(d, d, (c * c)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(c, Float64(b / d), Float64(-a)) / d)
	tmp = 0.0
	if (d <= -4.8e-27)
		tmp = t_0;
	elseif (d <= 8.5e-93)
		tmp = Float64(b / c);
	elseif (d <= 1.05e+69)
		tmp = Float64(Float64(-a) * Float64(d / 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[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -4.8e-27], t$95$0, If[LessEqual[d, 8.5e-93], N[(b / c), $MachinePrecision], If[LessEqual[d, 1.05e+69], N[((-a) * N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.80000000000000004e-27 or 1.05000000000000008e69 < d

   1. Initial program 48.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 48.7

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqr-abs-rev N/A

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

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

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

      12. rem-sqrt-square-rev N/A

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

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

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 48.7%

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

   5. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. div-add N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 86.4

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

   7. Applied rewrites 86.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}} \]
   8. Step-by-step derivation

   9. Applied rewrites 86.5%

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

   if -4.80000000000000004e-27 < d < 8.5000000000000007e-93

   1. Initial program 67.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 75.1

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

   5. Applied rewrites 75.1%

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

   if 8.5000000000000007e-93 < d < 1.05000000000000008e69

   1. Initial program 87.4%

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

   3. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 72.9

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

   5. Applied rewrites 72.9%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{+69}:\\ \;\;\;\;\left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -1.05 \cdot 10^{-21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{+69}:\\ \;\;\;\;\left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -1.05e-21)
     t_0
     (if (<= d 8.5e-93)
       (/ b c)
       (if (<= d 1.05e+69) (* (- a) (/ d (fma d d (* c c)))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -1.05e-21) {
		tmp = t_0;
	} else if (d <= 8.5e-93) {
		tmp = b / c;
	} else if (d <= 1.05e+69) {
		tmp = -a * (d / fma(d, d, (c * c)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -1.05e-21)
		tmp = t_0;
	elseif (d <= 8.5e-93)
		tmp = Float64(b / c);
	elseif (d <= 1.05e+69)
		tmp = Float64(Float64(-a) * Float64(d / 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[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -1.05e-21], t$95$0, If[LessEqual[d, 8.5e-93], N[(b / c), $MachinePrecision], If[LessEqual[d, 1.05e+69], N[((-a) * N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.05000000000000006e-21 or 1.05000000000000008e69 < d

   1. Initial program 47.7%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if -1.05000000000000006e-21 < d < 8.5000000000000007e-93

   1. Initial program 68.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 74.7

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

   5. Applied rewrites 74.7%

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

   if 8.5000000000000007e-93 < d < 1.05000000000000008e69

   1. Initial program 87.4%

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

   3. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 72.9

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

   5. Applied rewrites 72.9%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.05 \cdot 10^{-21}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{+69}:\\ \;\;\;\;\left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.05 \cdot 10^{-21} \lor \neg \left(d \leq 5.2 \cdot 10^{-92}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.05e-21) (not (<= d 5.2e-92))) (/ (- a) d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.05e-21) || !(d <= 5.2e-92)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-1.05d-21)) .or. (.not. (d <= 5.2d-92))) then
        tmp = -a / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.05e-21) || !(d <= 5.2e-92)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.05e-21) or not (d <= 5.2e-92):
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.05e-21) || !(d <= 5.2e-92))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.05e-21) || ~((d <= 5.2e-92)))
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.05e-21], N[Not[LessEqual[d, 5.2e-92]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.05000000000000006e-21 or 5.2e-92 < d

   1. Initial program 56.6%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 71.6

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

   5. Applied rewrites 71.6%

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

   if -1.05000000000000006e-21 < d < 5.2e-92

   1. Initial program 68.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 74.7

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

   5. Applied rewrites 74.7%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.05 \cdot 10^{-21} \lor \neg \left(d \leq 5.2 \cdot 10^{-92}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.3% 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 61.8%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 43.4

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

5. Applied rewrites 43.4%

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

6. Final simplification 43.4%

   \[\leadsto \frac{b}{c} \]
7. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.6× 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 2024354 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

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

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