Complex division, imag part

Percentage Accurate: 62.1% → 83.0%

Time: 3.8s

Alternatives: 10

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: 83.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{-a}{c}\\ \mathbf{if}\;c \leq -1.25 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \frac{d}{c}, \frac{b}{c}\right)\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{+51}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, d, b\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))) (t_1 (/ (- a) c)))
   (if (<= c -1.25e+116)
     (fma t_1 (/ d c) (/ b c))
     (if (<= c -5.5e-154)
       t_0
       (if (<= c 1.65e-143)
         (/ (- (/ (* c b) d) a) d)
         (if (<= c 6.2e+51) t_0 (/ (fma t_1 d b) c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = -a / c;
	double tmp;
	if (c <= -1.25e+116) {
		tmp = fma(t_1, (d / c), (b / c));
	} else if (c <= -5.5e-154) {
		tmp = t_0;
	} else if (c <= 1.65e-143) {
		tmp = (((c * b) / d) - a) / d;
	} else if (c <= 6.2e+51) {
		tmp = t_0;
	} else {
		tmp = fma(t_1, d, b) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(-a) / c)
	tmp = 0.0
	if (c <= -1.25e+116)
		tmp = fma(t_1, Float64(d / c), Float64(b / c));
	elseif (c <= -5.5e-154)
		tmp = t_0;
	elseif (c <= 1.65e-143)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	elseif (c <= 6.2e+51)
		tmp = t_0;
	else
		tmp = Float64(fma(t_1, d, b) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-a) / c), $MachinePrecision]}, If[LessEqual[c, -1.25e+116], N[(t$95$1 * N[(d / c), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -5.5e-154], t$95$0, If[LessEqual[c, 1.65e-143], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 6.2e+51], t$95$0, N[(N[(t$95$1 * d + b), $MachinePrecision] / c), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.25000000000000006e116

   1. Initial program 38.5%

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

   3. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if -1.25000000000000006e116 < c < -5.50000000000000002e-154 or 1.65e-143 < c < 6.20000000000000022e51

   1. Initial program 77.0%

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

   if -5.50000000000000002e-154 < c < 1.65e-143

   1. Initial program 68.5%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   4. 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 92.5

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 92.5

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

   8. Applied rewrites 92.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 92.7

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

   10. Applied rewrites 92.7%

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

   if 6.20000000000000022e51 < c

   1. Initial program 43.3%

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

   3. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 79.2

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

   5. Applied rewrites 79.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. div-add-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-frac-neg N/A

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

      14. lift-/.f64 N/A

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

      15. lift-neg.f64 79.9

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

   7. Applied rewrites 79.9%

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

Alternative 2: 82.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.25 \cdot 10^{+116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, \frac{d}{c}, b\right)}{c}\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{+51}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-a}{c}, d, b\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))))
   (if (<= c -1.25e+116)
     (/ (fma (- a) (/ d c) b) c)
     (if (<= c -5.5e-154)
       t_0
       (if (<= c 1.65e-143)
         (/ (- (/ (* c b) d) a) d)
         (if (<= c 6.2e+51) t_0 (/ (fma (/ (- a) c) d b) c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1.25e+116) {
		tmp = fma(-a, (d / c), b) / c;
	} else if (c <= -5.5e-154) {
		tmp = t_0;
	} else if (c <= 1.65e-143) {
		tmp = (((c * b) / d) - a) / d;
	} else if (c <= 6.2e+51) {
		tmp = t_0;
	} else {
		tmp = fma((-a / c), d, b) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -1.25e+116)
		tmp = Float64(fma(Float64(-a), Float64(d / c), b) / c);
	elseif (c <= -5.5e-154)
		tmp = t_0;
	elseif (c <= 1.65e-143)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	elseif (c <= 6.2e+51)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(Float64(-a) / c), d, b) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.25e+116], N[(N[((-a) * N[(d / c), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -5.5e-154], t$95$0, If[LessEqual[c, 1.65e-143], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 6.2e+51], t$95$0, N[(N[(N[((-a) / c), $MachinePrecision] * d + b), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.25000000000000006e116

   1. Initial program 38.5%

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

   3. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 84.7

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

   5. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-/.f64 84.6

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

   7. Applied rewrites 84.6%

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

   if -1.25000000000000006e116 < c < -5.50000000000000002e-154 or 1.65e-143 < c < 6.20000000000000022e51

   1. Initial program 77.0%

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

   if -5.50000000000000002e-154 < c < 1.65e-143

   1. Initial program 68.5%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   4. 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 92.5

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 92.5

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

   8. Applied rewrites 92.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 92.7

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

   10. Applied rewrites 92.7%

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

   if 6.20000000000000022e51 < c

   1. Initial program 43.3%

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

   3. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 79.2

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

   5. Applied rewrites 79.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. div-add-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-frac-neg N/A

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

      14. lift-/.f64 N/A

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

      15. lift-neg.f64 79.9

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

   7. Applied rewrites 79.9%

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

Alternative 3: 77.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.95 \cdot 10^{+98}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, \frac{d}{c}, b\right)}{c}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{c}{d} \cdot b - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-a}{c}, d, b\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.95e+98)
   (/ (fma (- a) (/ d c) b) c)
   (if (<= c 8e-28) (/ (- (* (/ c d) b) a) d) (/ (fma (/ (- a) c) d b) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.95e+98) {
		tmp = fma(-a, (d / c), b) / c;
	} else if (c <= 8e-28) {
		tmp = (((c / d) * b) - a) / d;
	} else {
		tmp = fma((-a / c), d, b) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.95e+98)
		tmp = Float64(fma(Float64(-a), Float64(d / c), b) / c);
	elseif (c <= 8e-28)
		tmp = Float64(Float64(Float64(Float64(c / d) * b) - a) / d);
	else
		tmp = Float64(fma(Float64(Float64(-a) / c), d, b) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.95e+98], N[(N[((-a) * N[(d / c), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 8e-28], N[(N[(N[(N[(c / d), $MachinePrecision] * b), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(N[((-a) / c), $MachinePrecision] * d + b), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.95e98

   1. Initial program 41.2%

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

   3. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 83.7

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

   5. Applied rewrites 83.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-/.f64 83.5

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

   7. Applied rewrites 83.5%

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

   if -1.95e98 < c < 7.99999999999999977e-28

   1. Initial program 73.1%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   4. 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 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 77.3

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

   8. Applied rewrites 77.3%

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

   if 7.99999999999999977e-28 < c

   1. Initial program 51.3%

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

   3. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 73.0

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

   5. Applied rewrites 73.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. div-add-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-frac-neg N/A

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

      14. lift-/.f64 N/A

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

      15. lift-neg.f64 73.5

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

   7. Applied rewrites 73.5%

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

Alternative 4: 77.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(-a, \frac{d}{c}, b\right)}{c}\\ \mathbf{if}\;c \leq -1.95 \cdot 10^{+98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{c}{d} \cdot b - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (- a) (/ d c) b) c)))
   (if (<= c -1.95e+98) t_0 (if (<= c 8e-28) (/ (- (* (/ c d) b) a) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(-a, (d / c), b) / c;
	double tmp;
	if (c <= -1.95e+98) {
		tmp = t_0;
	} else if (c <= 8e-28) {
		tmp = (((c / d) * b) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(-a), Float64(d / c), b) / c)
	tmp = 0.0
	if (c <= -1.95e+98)
		tmp = t_0;
	elseif (c <= 8e-28)
		tmp = Float64(Float64(Float64(Float64(c / d) * b) - a) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[((-a) * N[(d / c), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -1.95e+98], t$95$0, If[LessEqual[c, 8e-28], N[(N[(N[(N[(c / d), $MachinePrecision] * b), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.95e98 or 7.99999999999999977e-28 < c

   1. Initial program 47.5%

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

   3. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 77.1

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

   5. Applied rewrites 77.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-/.f64 76.7

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

   7. Applied rewrites 76.7%

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

   if -1.95e98 < c < 7.99999999999999977e-28

   1. Initial program 73.1%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   4. 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 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 77.3

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

   8. Applied rewrites 77.3%

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

Alternative 5: 77.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{d} \cdot b - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.1e-5)
   (/ (- (/ (* c b) d) a) d)
   (if (<= d 3.6e-15) (/ (- b (/ (* d a) c)) c) (/ (- (* (/ c d) b) a) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.1e-5) {
		tmp = (((c * b) / d) - a) / d;
	} else if (d <= 3.6e-15) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = (((c / d) * b) - a) / d;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-1.1d-5)) then
        tmp = (((c * b) / d) - a) / d
    else if (d <= 3.6d-15) then
        tmp = (b - ((d * a) / c)) / c
    else
        tmp = (((c / d) * b) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.1e-5) {
		tmp = (((c * b) / d) - a) / d;
	} else if (d <= 3.6e-15) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = (((c / d) * b) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.1e-5:
		tmp = (((c * b) / d) - a) / d
	elif d <= 3.6e-15:
		tmp = (b - ((d * a) / c)) / c
	else:
		tmp = (((c / d) * b) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.1e-5)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	elseif (d <= 3.6e-15)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	else
		tmp = Float64(Float64(Float64(Float64(c / d) * b) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -1.1e-5)
		tmp = (((c * b) / d) - a) / d;
	elseif (d <= 3.6e-15)
		tmp = (b - ((d * a) / c)) / c;
	else
		tmp = (((c / d) * b) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.1e-5], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 3.6e-15], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(N[(c / d), $MachinePrecision] * b), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.1e-5

   1. Initial program 51.6%

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

   3. Taylor expanded in d around inf

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

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 77.7

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

   8. Applied rewrites 77.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.8

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

   10. Applied rewrites 73.8%

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

   if -1.1e-5 < d < 3.6000000000000001e-15

   1. Initial program 74.7%

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

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 80.5

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

   5. Applied rewrites 80.5%

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

   if 3.6000000000000001e-15 < d

   1. Initial program 49.0%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 75.4

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

   8. Applied rewrites 75.4%

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

Alternative 6: 72.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.2 \cdot 10^{+58}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.38 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -4.2e+58)
   (/ b c)
   (if (<= c 1.38e-27) (/ (- (/ (* c b) d) a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.2e+58) {
		tmp = b / c;
	} else if (c <= 1.38e-27) {
		tmp = (((c * b) / d) - a) / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-4.2d+58)) then
        tmp = b / c
    else if (c <= 1.38d-27) then
        tmp = (((c * b) / d) - a) / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.2e+58) {
		tmp = b / c;
	} else if (c <= 1.38e-27) {
		tmp = (((c * b) / d) - a) / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -4.2e+58:
		tmp = b / c
	elif c <= 1.38e-27:
		tmp = (((c * b) / d) - a) / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -4.2e+58)
		tmp = Float64(b / c);
	elseif (c <= 1.38e-27)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -4.2e+58)
		tmp = b / c;
	elseif (c <= 1.38e-27)
		tmp = (((c * b) / d) - a) / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -4.2e+58], N[(b / c), $MachinePrecision], If[LessEqual[c, 1.38e-27], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -4.20000000000000024e58 or 1.38e-27 < c

   1. Initial program 49.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 65.1

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

   5. Applied rewrites 65.1%

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

   if -4.20000000000000024e58 < c < 1.38e-27

   1. Initial program 73.3%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   4. 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 79.4

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

   5. Applied rewrites 79.4%

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

   6. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 79.4

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

   8. Applied rewrites 79.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 79.3

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

   10. Applied rewrites 79.3%

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

Alternative 7: 72.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.1 \cdot 10^{+98}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.38 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{c}{d} \cdot b - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.1e+98)
   (/ b c)
   (if (<= c 1.38e-27) (/ (- (* (/ c d) b) a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.1e+98) {
		tmp = b / c;
	} else if (c <= 1.38e-27) {
		tmp = (((c / d) * b) - a) / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-3.1d+98)) then
        tmp = b / c
    else if (c <= 1.38d-27) then
        tmp = (((c / d) * b) - a) / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.1e+98) {
		tmp = b / c;
	} else if (c <= 1.38e-27) {
		tmp = (((c / d) * b) - a) / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -3.1e+98:
		tmp = b / c
	elif c <= 1.38e-27:
		tmp = (((c / d) * b) - a) / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.1e+98)
		tmp = Float64(b / c);
	elseif (c <= 1.38e-27)
		tmp = Float64(Float64(Float64(Float64(c / d) * b) - a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -3.1e+98)
		tmp = b / c;
	elseif (c <= 1.38e-27)
		tmp = (((c / d) * b) - a) / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.1e+98], N[(b / c), $MachinePrecision], If[LessEqual[c, 1.38e-27], N[(N[(N[(N[(c / d), $MachinePrecision] * b), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -3.10000000000000019e98 or 1.38e-27 < c

   1. Initial program 47.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 66.4

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

   5. Applied rewrites 66.4%

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

   if -3.10000000000000019e98 < c < 1.38e-27

   1. Initial program 73.1%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   4. 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 77.2

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

   5. Applied rewrites 77.2%

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

   6. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 77.2

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

   8. Applied rewrites 77.2%

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

Alternative 8: 65.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -3.2 \cdot 10^{-119}:\\ \;\;\;\;b \cdot \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-28}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -4.5e+143)
   (/ b c)
   (if (<= c -3.2e-119)
     (* b (/ c (fma d d (* c c))))
     (if (<= c 8e-28) (/ (- a) d) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.5e+143) {
		tmp = b / c;
	} else if (c <= -3.2e-119) {
		tmp = b * (c / fma(d, d, (c * c)));
	} else if (c <= 8e-28) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -4.5e+143)
		tmp = Float64(b / c);
	elseif (c <= -3.2e-119)
		tmp = Float64(b * Float64(c / fma(d, d, Float64(c * c))));
	elseif (c <= 8e-28)
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -4.5e+143], N[(b / c), $MachinePrecision], If[LessEqual[c, -3.2e-119], N[(b * N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 8e-28], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -4.4999999999999997e143 or 7.99999999999999977e-28 < c

   1. Initial program 46.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 66.9

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

   5. Applied rewrites 66.9%

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

   if -4.4999999999999997e143 < c < -3.19999999999999993e-119

   1. Initial program 74.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{b \cdot c}{{c}^{2} + {d}^{2}}} \]
   4. 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 55.1

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

   5. Applied rewrites 55.1%

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

   if -3.19999999999999993e-119 < c < 7.99999999999999977e-28

   1. Initial program 71.4%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 69.8

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

   5. Applied rewrites 69.8%

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

Alternative 9: 64.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{+58}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-28}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.1e+58) (/ b c) (if (<= c 8e-28) (/ (- a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.1e+58) {
		tmp = b / c;
	} else if (c <= 8e-28) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.1d+58)) then
        tmp = b / c
    else if (c <= 8d-28) then
        tmp = -a / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.1e+58) {
		tmp = b / c;
	} else if (c <= 8e-28) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.1e+58:
		tmp = b / c
	elif c <= 8e-28:
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.1e+58)
		tmp = Float64(b / c);
	elseif (c <= 8e-28)
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.1e+58)
		tmp = b / c;
	elseif (c <= 8e-28)
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.1e+58], N[(b / c), $MachinePrecision], If[LessEqual[c, 8e-28], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.1e58 or 7.99999999999999977e-28 < c

   1. Initial program 49.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 65.1

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

   5. Applied rewrites 65.1%

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

   if -1.1e58 < c < 7.99999999999999977e-28

   1. Initial program 73.3%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 63.2

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

   5. Applied rewrites 63.2%

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

Alternative 10: 41.7% 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 62.1%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 41.7

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

5. Applied rewrites 41.7%

   \[\leadsto \color{blue}{\frac{b}{c}} \]
6. 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 2025087 
(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))))