Complex division, imag part

Percentage Accurate: 62.4% → 83.3%

Time: 3.2s

Alternatives: 10

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ t_1 := \frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{if}\;d \leq -4.8 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-123}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{c}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+103}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (- a) d (* c b)) (fma d d (* c c))))
        (t_1 (/ (fma (/ b d) c (- a)) d)))
   (if (<= d -4.8e+80)
     t_1
     (if (<= d -2.6e-156)
       t_0
       (if (<= d 2.4e-123)
         (- (/ (fma a (/ d c) (- b)) c))
         (if (<= d 2e+103) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(-a, d, (c * b)) / fma(d, d, (c * c));
	double t_1 = fma((b / d), c, -a) / d;
	double tmp;
	if (d <= -4.8e+80) {
		tmp = t_1;
	} else if (d <= -2.6e-156) {
		tmp = t_0;
	} else if (d <= 2.4e-123) {
		tmp = -(fma(a, (d / c), -b) / c);
	} else if (d <= 2e+103) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(-a), d, Float64(c * b)) / fma(d, d, Float64(c * c)))
	t_1 = Float64(fma(Float64(b / d), c, Float64(-a)) / d)
	tmp = 0.0
	if (d <= -4.8e+80)
		tmp = t_1;
	elseif (d <= -2.6e-156)
		tmp = t_0;
	elseif (d <= 2.4e-123)
		tmp = Float64(-Float64(fma(a, Float64(d / c), Float64(-b)) / c));
	elseif (d <= 2e+103)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[((-a) * d + N[(c * b), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b / d), $MachinePrecision] * c + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -4.8e+80], t$95$1, If[LessEqual[d, -2.6e-156], t$95$0, If[LessEqual[d, 2.4e-123], (-N[(N[(a * N[(d / c), $MachinePrecision] + (-b)), $MachinePrecision] / c), $MachinePrecision]), If[LessEqual[d, 2e+103], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.79999999999999958e80 or 2e103 < d

   1. Initial program 39.0%

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

   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 82.5

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

   4. Applied rewrites 82.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. div-add N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r/ N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-neg.f64 83.4

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

   6. Applied rewrites 83.4%

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

   if -4.79999999999999958e80 < d < -2.6000000000000001e-156 or 2.4e-123 < d < 2e103

   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

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.8

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. pow2 N/A

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

      19. +-commutative N/A

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

      20. pow2 N/A

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

      21. lower-fma.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 76.8

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

   3. Applied rewrites 76.8%

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

   if -2.6000000000000001e-156 < d < 2.4e-123

   1. Initial program 72.0%

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

   2. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 92.0

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

   4. Applied rewrites 92.0%

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

Alternative 2: 72.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -7.4 \cdot 10^{+124}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.4 \cdot 10^{-31}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{d \cdot d}\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{-69}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{+139}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, b, \left(-a\right) \cdot d\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -7.4e+124)
     t_0
     (if (<= d -2.4e-31)
       (/ (- (* b c) (* a d)) (* d d))
       (if (<= d 1.3e-69)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 1.65e+139) (/ (fma c b (* (- a) d)) (* d d)) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -7.4e+124) {
		tmp = t_0;
	} else if (d <= -2.4e-31) {
		tmp = ((b * c) - (a * d)) / (d * d);
	} else if (d <= 1.3e-69) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 1.65e+139) {
		tmp = fma(c, b, (-a * d)) / (d * d);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -7.4e+124)
		tmp = t_0;
	elseif (d <= -2.4e-31)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(d * d));
	elseif (d <= 1.3e-69)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 1.65e+139)
		tmp = Float64(fma(c, b, Float64(Float64(-a) * d)) / Float64(d * d));
	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, -7.4e+124], t$95$0, If[LessEqual[d, -2.4e-31], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.3e-69], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.65e+139], N[(N[(c * b + N[((-a) * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7.40000000000000016e124 or 1.6500000000000001e139 < d

   1. Initial program 32.1%

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

   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 75.6

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

   4. Applied rewrites 75.6%

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

   if -7.40000000000000016e124 < d < -2.4e-31

   1. Initial program 75.4%

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

   2. Taylor expanded in c around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 56.2

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

   4. Applied rewrites 56.2%

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

   if -2.4e-31 < d < 1.3000000000000001e-69

   1. Initial program 74.6%

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

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   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 85.2

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

   4. Applied rewrites 85.2%

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

   if 1.3000000000000001e-69 < d < 1.6500000000000001e139

   1. Initial program 71.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 71.7

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. pow2 N/A

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

      19. +-commutative N/A

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

      20. pow2 N/A

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

      21. lower-fma.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 71.7

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

   3. Applied rewrites 71.7%

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

   4. Taylor expanded in c around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 49.1

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

   6. Applied rewrites 49.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-neg.f64 49.1

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

   8. Applied rewrites 49.1%

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

Alternative 3: 65.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -7.4 \cdot 10^{+124}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.4 \cdot 10^{-31}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{d \cdot d}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-79}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{+139}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, b, \left(-a\right) \cdot d\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -7.4e+124)
     t_0
     (if (<= d -2.4e-31)
       (/ (- (* b c) (* a d)) (* d d))
       (if (<= d 3.6e-79)
         (/ b c)
         (if (<= d 1.65e+139) (/ (fma c b (* (- a) d)) (* d d)) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -7.4e+124) {
		tmp = t_0;
	} else if (d <= -2.4e-31) {
		tmp = ((b * c) - (a * d)) / (d * d);
	} else if (d <= 3.6e-79) {
		tmp = b / c;
	} else if (d <= 1.65e+139) {
		tmp = fma(c, b, (-a * d)) / (d * d);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -7.4e+124)
		tmp = t_0;
	elseif (d <= -2.4e-31)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(d * d));
	elseif (d <= 3.6e-79)
		tmp = Float64(b / c);
	elseif (d <= 1.65e+139)
		tmp = Float64(fma(c, b, Float64(Float64(-a) * d)) / Float64(d * d));
	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, -7.4e+124], t$95$0, If[LessEqual[d, -2.4e-31], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.6e-79], N[(b / c), $MachinePrecision], If[LessEqual[d, 1.65e+139], N[(N[(c * b + N[((-a) * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7.40000000000000016e124 or 1.6500000000000001e139 < d

   1. Initial program 32.1%

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

   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 75.6

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

   4. Applied rewrites 75.6%

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

   if -7.40000000000000016e124 < d < -2.4e-31

   1. Initial program 75.4%

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

   2. Taylor expanded in c around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 56.2

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

   4. Applied rewrites 56.2%

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

   if -2.4e-31 < d < 3.6000000000000002e-79

   1. Initial program 74.4%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 68.3

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

   4. Applied rewrites 68.3%

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

   if 3.6000000000000002e-79 < d < 1.6500000000000001e139

   1. Initial program 72.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 72.3

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. pow2 N/A

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

      19. +-commutative N/A

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

      20. pow2 N/A

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

      21. lower-fma.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 72.3

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

   3. Applied rewrites 72.3%

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

   4. Taylor expanded in c around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 49.2

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

   6. Applied rewrites 49.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-neg.f64 49.2

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

   8. Applied rewrites 49.2%

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

Alternative 4: 65.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ t_1 := \frac{b \cdot c - a \cdot d}{d \cdot d}\\ \mathbf{if}\;d \leq -7.4 \cdot 10^{+124}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.4 \cdot 10^{-31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-79}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{+139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)) (t_1 (/ (- (* b c) (* a d)) (* d d))))
   (if (<= d -7.4e+124)
     t_0
     (if (<= d -2.4e-31)
       t_1
       (if (<= d 3.6e-79) (/ b c) (if (<= d 1.65e+139) t_1 t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double t_1 = ((b * c) - (a * d)) / (d * d);
	double tmp;
	if (d <= -7.4e+124) {
		tmp = t_0;
	} else if (d <= -2.4e-31) {
		tmp = t_1;
	} else if (d <= 3.6e-79) {
		tmp = b / c;
	} else if (d <= 1.65e+139) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -a / d
    t_1 = ((b * c) - (a * d)) / (d * d)
    if (d <= (-7.4d+124)) then
        tmp = t_0
    else if (d <= (-2.4d-31)) then
        tmp = t_1
    else if (d <= 3.6d-79) then
        tmp = b / c
    else if (d <= 1.65d+139) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double t_1 = ((b * c) - (a * d)) / (d * d);
	double tmp;
	if (d <= -7.4e+124) {
		tmp = t_0;
	} else if (d <= -2.4e-31) {
		tmp = t_1;
	} else if (d <= 3.6e-79) {
		tmp = b / c;
	} else if (d <= 1.65e+139) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	t_1 = ((b * c) - (a * d)) / (d * d)
	tmp = 0
	if d <= -7.4e+124:
		tmp = t_0
	elif d <= -2.4e-31:
		tmp = t_1
	elif d <= 3.6e-79:
		tmp = b / c
	elif d <= 1.65e+139:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	t_1 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(d * d))
	tmp = 0.0
	if (d <= -7.4e+124)
		tmp = t_0;
	elseif (d <= -2.4e-31)
		tmp = t_1;
	elseif (d <= 3.6e-79)
		tmp = Float64(b / c);
	elseif (d <= 1.65e+139)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -a / d;
	t_1 = ((b * c) - (a * d)) / (d * d);
	tmp = 0.0;
	if (d <= -7.4e+124)
		tmp = t_0;
	elseif (d <= -2.4e-31)
		tmp = t_1;
	elseif (d <= 3.6e-79)
		tmp = b / c;
	elseif (d <= 1.65e+139)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -7.4e+124], t$95$0, If[LessEqual[d, -2.4e-31], t$95$1, If[LessEqual[d, 3.6e-79], N[(b / c), $MachinePrecision], If[LessEqual[d, 1.65e+139], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.40000000000000016e124 or 1.6500000000000001e139 < d

   1. Initial program 32.1%

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

   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 75.6

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

   4. Applied rewrites 75.6%

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

   if -7.40000000000000016e124 < d < -2.4e-31 or 3.6000000000000002e-79 < d < 1.6500000000000001e139

   1. Initial program 73.6%

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

   2. Taylor expanded in c around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 52.1

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

   4. Applied rewrites 52.1%

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

   if -2.4e-31 < d < 3.6000000000000002e-79

   1. Initial program 74.4%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 68.3

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

   4. Applied rewrites 68.3%

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

Alternative 5: 64.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.85 \cdot 10^{-8}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.2 \cdot 10^{-164}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{+111}:\\ \;\;\;\;b \cdot \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.85e-8)
   (/ b c)
   (if (<= c 1.2e-164)
     (/ (- a) d)
     (if (<= c 2.2e+111) (* b (/ c (fma d d (* c c)))) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.85e-8) {
		tmp = b / c;
	} else if (c <= 1.2e-164) {
		tmp = -a / d;
	} else if (c <= 2.2e+111) {
		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 <= -5.85e-8)
		tmp = Float64(b / c);
	elseif (c <= 1.2e-164)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 2.2e+111)
		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, -5.85e-8], N[(b / c), $MachinePrecision], If[LessEqual[c, 1.2e-164], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 2.2e+111], 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 < -5.8500000000000001e-8 or 2.19999999999999999e111 < c

   1. Initial program 45.9%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 68.0

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

   4. Applied rewrites 68.0%

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

   if -5.8500000000000001e-8 < c < 1.19999999999999992e-164

   1. Initial program 72.3%

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

   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 67.6

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

   4. Applied rewrites 67.6%

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

   if 1.19999999999999992e-164 < c < 2.19999999999999999e111

   1. Initial program 76.6%

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

   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 53.7

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

   4. Applied rewrites 53.7%

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

Alternative 6: 77.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{if}\;d \leq -2.4 \cdot 10^{-31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-66}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ b d) c (- a)) d)))
   (if (<= d -2.4e-31)
     t_0
     (if (<= d 1.45e-66) (- (/ (fma a (/ d c) (- b)) c)) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((b / d), c, -a) / d;
	double tmp;
	if (d <= -2.4e-31) {
		tmp = t_0;
	} else if (d <= 1.45e-66) {
		tmp = -(fma(a, (d / c), -b) / c);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(b / d), c, Float64(-a)) / d)
	tmp = 0.0
	if (d <= -2.4e-31)
		tmp = t_0;
	elseif (d <= 1.45e-66)
		tmp = Float64(-Float64(fma(a, Float64(d / c), Float64(-b)) / c));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b / d), $MachinePrecision] * c + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2.4e-31], t$95$0, If[LessEqual[d, 1.45e-66], (-N[(N[(a * N[(d / c), $MachinePrecision] + (-b)), $MachinePrecision] / c), $MachinePrecision]), t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -2.4e-31 or 1.45000000000000006e-66 < d

   1. Initial program 53.2%

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

   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 71.3

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

   4. Applied rewrites 71.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. div-add N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r/ N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-neg.f64 71.8

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

   6. Applied rewrites 71.8%

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

   if -2.4e-31 < d < 1.45000000000000006e-66

   1. Initial program 74.6%

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

   2. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 85.2

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

   4. Applied rewrites 85.2%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{c}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 77.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{if}\;d \leq -2.4 \cdot 10^{-31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-66}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ b d) c (- a)) d)))
   (if (<= d -2.4e-31)
     t_0
     (if (<= d 1.45e-66) (/ (- b (/ (* d a) c)) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((b / d), c, -a) / d;
	double tmp;
	if (d <= -2.4e-31) {
		tmp = t_0;
	} else if (d <= 1.45e-66) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(b / d), c, Float64(-a)) / d)
	tmp = 0.0
	if (d <= -2.4e-31)
		tmp = t_0;
	elseif (d <= 1.45e-66)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b / d), $MachinePrecision] * c + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2.4e-31], t$95$0, If[LessEqual[d, 1.45e-66], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -2.4e-31 or 1.45000000000000006e-66 < d

   1. Initial program 53.2%

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

   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 71.3

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

   4. Applied rewrites 71.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. div-add N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r/ N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-neg.f64 71.8

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

   6. Applied rewrites 71.8%

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

   if -2.4e-31 < d < 1.45000000000000006e-66

   1. Initial program 74.6%

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

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   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 85.2

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

   4. Applied rewrites 85.2%

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

Alternative 8: 77.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -2.4 \cdot 10^{-31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-71}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma b (/ c d) (- a)) d)))
   (if (<= d -2.4e-31) t_0 (if (<= d 3.3e-71) (/ (- b (/ (* d a) c)) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(b, (c / d), -a) / d;
	double tmp;
	if (d <= -2.4e-31) {
		tmp = t_0;
	} else if (d <= 3.3e-71) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(b, Float64(c / d), Float64(-a)) / d)
	tmp = 0.0
	if (d <= -2.4e-31)
		tmp = t_0;
	elseif (d <= 3.3e-71)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2.4e-31], t$95$0, If[LessEqual[d, 3.3e-71], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -2.4e-31 or 3.3000000000000002e-71 < d

   1. Initial program 53.4%

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

   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 71.0

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

   4. Applied rewrites 71.0%

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

   if -2.4e-31 < d < 3.3000000000000002e-71

   1. Initial program 74.6%

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

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   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 85.3

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

   4. Applied rewrites 85.3%

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

Alternative 9: 61.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.85 \cdot 10^{-8}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 3 \cdot 10^{-150}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.85e-8) (/ b c) (if (<= c 3e-150) (/ (- a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.85e-8) {
		tmp = b / c;
	} else if (c <= 3e-150) {
		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 <= (-5.85d-8)) then
        tmp = b / c
    else if (c <= 3d-150) 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 <= -5.85e-8) {
		tmp = b / c;
	} else if (c <= 3e-150) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -5.85e-8:
		tmp = b / c
	elif c <= 3e-150:
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.85e-8)
		tmp = Float64(b / c);
	elseif (c <= 3e-150)
		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 <= -5.85e-8)
		tmp = b / c;
	elseif (c <= 3e-150)
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.85e-8], N[(b / c), $MachinePrecision], If[LessEqual[c, 3e-150], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -5.8500000000000001e-8 or 3.0000000000000002e-150 < c

   1. Initial program 56.5%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 57.9

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

   4. Applied rewrites 57.9%

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

   if -5.8500000000000001e-8 < c < 3.0000000000000002e-150

   1. Initial program 72.3%

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

   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 67.6

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

   4. Applied rewrites 67.6%

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

Alternative 10: 43.6% 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 62.4%

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

2. Taylor expanded in c around inf

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

   1. lower-/.f64 43.6

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

4. Applied rewrites 43.6%

   \[\leadsto \color{blue}{\frac{b}{c}} \]
5. 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 2025105 
(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))))