Complex division, imag part

Percentage Accurate: 61.9% → 83.1%

Time: 9.3s

Alternatives: 12

Speedup: 1.1×

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.9% 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

real(8) function code(a, b, c, d)
    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: 83.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -4.15 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{d}{\frac{c}{a}} - b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -4 \cdot 10^{-46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+75}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))))
   (if (<= c -4.15e+80)
     (/ (- (/ d (/ c a)) b) (hypot c d))
     (if (<= c -4e-46)
       t_0
       (if (<= c 4.5e-110)
         (/ (- (/ (* b c) d) a) d)
         (if (<= c 8e+75) t_0 (/ (- b (* a (/ d c))) 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 <= -4.15e+80) {
		tmp = ((d / (c / a)) - b) / hypot(c, d);
	} else if (c <= -4e-46) {
		tmp = t_0;
	} else if (c <= 4.5e-110) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 8e+75) {
		tmp = t_0;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Java

public static 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 <= -4.15e+80) {
		tmp = ((d / (c / a)) - b) / Math.hypot(c, d);
	} else if (c <= -4e-46) {
		tmp = t_0;
	} else if (c <= 4.5e-110) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 8e+75) {
		tmp = t_0;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -4.15e+80:
		tmp = ((d / (c / a)) - b) / math.hypot(c, d)
	elif c <= -4e-46:
		tmp = t_0
	elif c <= 4.5e-110:
		tmp = (((b * c) / d) - a) / d
	elif c <= 8e+75:
		tmp = t_0
	else:
		tmp = (b - (a * (d / c))) / 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 <= -4.15e+80)
		tmp = Float64(Float64(Float64(d / Float64(c / a)) - b) / hypot(c, d));
	elseif (c <= -4e-46)
		tmp = t_0;
	elseif (c <= 4.5e-110)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 8e+75)
		tmp = t_0;
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -4.15e+80)
		tmp = ((d / (c / a)) - b) / hypot(c, d);
	elseif (c <= -4e-46)
		tmp = t_0;
	elseif (c <= 4.5e-110)
		tmp = (((b * c) / d) - a) / d;
	elseif (c <= 8e+75)
		tmp = t_0;
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = 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, -4.15e+80], N[(N[(N[(d / N[(c / a), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -4e-46], t$95$0, If[LessEqual[c, 4.5e-110], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 8e+75], t$95$0, N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -4.15000000000000002e80

   1. Initial program 28.5%

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

      1. *-un-lft-identity 28.5%

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

      2. add-sqr-sqrt 28.5%

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

      3. times-frac 28.5%

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

      4. hypot-define 28.5%

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

      5. fma-neg 28.5%

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

      6. distribute-rgt-neg-in 28.5%

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

      7. hypot-define 61.2%

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

   4. Applied egg-rr 61.2%

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

   5. Taylor expanded in c around -inf 82.7%

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

      1. associate-*r/ 80.7%

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

      2. +-commutative 80.7%

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

      3. mul-1-neg 80.7%

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

      4. unsub-neg 80.7%

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

      5. *-commutative 80.7%

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

      6. associate-*l/ 82.7%

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

      7. associate-*r/ 82.7%

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

   7. Simplified 82.7%

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

      1. associate-*l/ 83.0%

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

      2. *-un-lft-identity 83.0%

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

      3. clear-num 83.0%

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

      4. un-div-inv 83.0%

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

   9. Applied egg-rr 83.0%

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

   if -4.15000000000000002e80 < c < -4.00000000000000009e-46 or 4.5000000000000001e-110 < c < 7.99999999999999941e75

   1. Initial program 87.8%

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

   if -4.00000000000000009e-46 < c < 4.5000000000000001e-110

   1. Initial program 66.3%

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

   3. Taylor expanded in d around inf 89.2%

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

      1. +-commutative 89.2%

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

      2. mul-1-neg 89.2%

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

      3. unsub-neg 89.2%

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

   5. Applied egg-rr 89.2%

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

   if 7.99999999999999941e75 < c

   1. Initial program 27.9%

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

   3. Taylor expanded in c around inf 72.9%

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

      1. mul-1-neg 72.9%

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

      2. unsub-neg 72.9%

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

      3. associate-/l* 83.6%

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

   5. Simplified 83.6%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.15 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{d}{\frac{c}{a}} - b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -4 \cdot 10^{-46}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+75}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+238}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(b, c, a \cdot \left(-d\right)\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) 2e+238)
   (* (/ 1.0 (hypot c d)) (/ (fma b c (* a (- d))) (hypot c d)))
   (/ (- b (* a (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((b * c) - (a * d)) / ((c * c) + (d * d))) <= 2e+238) {
		tmp = (1.0 / hypot(c, d)) * (fma(b, c, (a * -d)) / hypot(c, d));
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d))) <= 2e+238)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(b, c, Float64(a * Float64(-d))) / hypot(c, d)));
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+238], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(b * c + N[(a * (-d)), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 2.0000000000000001e238

   1. Initial program 76.9%

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

      1. *-un-lft-identity 76.9%

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

      2. add-sqr-sqrt 76.9%

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

      3. times-frac 76.9%

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

      4. hypot-define 76.9%

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

      5. fma-neg 76.9%

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

      6. distribute-rgt-neg-in 76.9%

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

      7. hypot-define 97.0%

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

   4. Applied egg-rr 97.0%

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

   if 2.0000000000000001e238 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 12.7%

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

   3. Taylor expanded in c around inf 51.6%

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

      1. mul-1-neg 51.6%

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

      2. unsub-neg 51.6%

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

      3. associate-/l* 59.6%

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

   5. Simplified 59.6%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+238}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(b, c, a \cdot \left(-d\right)\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -4.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;c \leq -3.5 \cdot 10^{-45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 2.35 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{+75}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))))
   (if (<= c -4.2e+97)
     (/ (- b (/ (* a d) c)) c)
     (if (<= c -3.5e-45)
       t_0
       (if (<= c 2.35e-110)
         (/ (- (/ (* b c) d) a) d)
         (if (<= c 9.5e+75) t_0 (/ (- b (* a (/ d c))) 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 <= -4.2e+97) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (c <= -3.5e-45) {
		tmp = t_0;
	} else if (c <= 2.35e-110) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 9.5e+75) {
		tmp = t_0;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
    if (c <= (-4.2d+97)) then
        tmp = (b - ((a * d) / c)) / c
    else if (c <= (-3.5d-45)) then
        tmp = t_0
    else if (c <= 2.35d-110) then
        tmp = (((b * c) / d) - a) / d
    else if (c <= 9.5d+75) then
        tmp = t_0
    else
        tmp = (b - (a * (d / c))) / c
    end if
    code = tmp
end function

Java

public static 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 <= -4.2e+97) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (c <= -3.5e-45) {
		tmp = t_0;
	} else if (c <= 2.35e-110) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 9.5e+75) {
		tmp = t_0;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -4.2e+97:
		tmp = (b - ((a * d) / c)) / c
	elif c <= -3.5e-45:
		tmp = t_0
	elif c <= 2.35e-110:
		tmp = (((b * c) / d) - a) / d
	elif c <= 9.5e+75:
		tmp = t_0
	else:
		tmp = (b - (a * (d / c))) / 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 <= -4.2e+97)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (c <= -3.5e-45)
		tmp = t_0;
	elseif (c <= 2.35e-110)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 9.5e+75)
		tmp = t_0;
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -4.2e+97)
		tmp = (b - ((a * d) / c)) / c;
	elseif (c <= -3.5e-45)
		tmp = t_0;
	elseif (c <= 2.35e-110)
		tmp = (((b * c) / d) - a) / d;
	elseif (c <= 9.5e+75)
		tmp = t_0;
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = 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, -4.2e+97], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -3.5e-45], t$95$0, If[LessEqual[c, 2.35e-110], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 9.5e+75], t$95$0, N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -4.20000000000000023e97

   1. Initial program 27.8%

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

   3. Taylor expanded in c around inf 86.3%

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

      1. mul-1-neg 86.3%

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

      2. unsub-neg 86.3%

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

      3. associate-/l* 84.0%

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

   5. Simplified 84.0%

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

   6. Taylor expanded in a around 0 86.3%

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

   if -4.20000000000000023e97 < c < -3.5e-45 or 2.34999999999999996e-110 < c < 9.50000000000000061e75

   1. Initial program 85.6%

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

   if -3.5e-45 < c < 2.34999999999999996e-110

   1. Initial program 66.3%

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

   3. Taylor expanded in d around inf 89.2%

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

      1. +-commutative 89.2%

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

      2. mul-1-neg 89.2%

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

      3. unsub-neg 89.2%

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

   5. Applied egg-rr 89.2%

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

   if 9.50000000000000061e75 < c

   1. Initial program 27.9%

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

   3. Taylor expanded in c around inf 72.9%

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

      1. mul-1-neg 72.9%

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

      2. unsub-neg 72.9%

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

      3. associate-/l* 83.6%

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

   5. Simplified 83.6%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;c \leq -3.5 \cdot 10^{-45}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.35 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;c \leq 4.3 \cdot 10^{-66} \lor \neg \left(c \leq 1.05 \cdot 10^{+69}\right) \land c \leq 9 \cdot 10^{+117}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -7.5e-25)
   (/ (- b (/ (* a d) c)) c)
   (if (or (<= c 4.3e-66) (and (not (<= c 1.05e+69)) (<= c 9e+117)))
     (/ (- (* b (/ c d)) a) d)
     (/ (- b (* a (/ d c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7.5e-25) {
		tmp = (b - ((a * d) / c)) / c;
	} else if ((c <= 4.3e-66) || (!(c <= 1.05e+69) && (c <= 9e+117))) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    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 <= (-7.5d-25)) then
        tmp = (b - ((a * d) / c)) / c
    else if ((c <= 4.3d-66) .or. (.not. (c <= 1.05d+69)) .and. (c <= 9d+117)) then
        tmp = ((b * (c / d)) - a) / d
    else
        tmp = (b - (a * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7.5e-25) {
		tmp = (b - ((a * d) / c)) / c;
	} else if ((c <= 4.3e-66) || (!(c <= 1.05e+69) && (c <= 9e+117))) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -7.5e-25:
		tmp = (b - ((a * d) / c)) / c
	elif (c <= 4.3e-66) or (not (c <= 1.05e+69) and (c <= 9e+117)):
		tmp = ((b * (c / d)) - a) / d
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -7.5e-25)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif ((c <= 4.3e-66) || (!(c <= 1.05e+69) && (c <= 9e+117)))
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -7.5e-25)
		tmp = (b - ((a * d) / c)) / c;
	elseif ((c <= 4.3e-66) || (~((c <= 1.05e+69)) && (c <= 9e+117)))
		tmp = ((b * (c / d)) - a) / d;
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -7.5e-25], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[Or[LessEqual[c, 4.3e-66], And[N[Not[LessEqual[c, 1.05e+69]], $MachinePrecision], LessEqual[c, 9e+117]]], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.49999999999999989e-25

   1. Initial program 50.3%

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

   3. Taylor expanded in c around inf 79.2%

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

      1. mul-1-neg 79.2%

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

      2. unsub-neg 79.2%

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

      3. associate-/l* 78.0%

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

   5. Simplified 78.0%

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

   6. Taylor expanded in a around 0 79.2%

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

   if -7.49999999999999989e-25 < c < 4.30000000000000013e-66 or 1.05000000000000008e69 < c < 9e117

   1. Initial program 68.2%

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

   3. Taylor expanded in c around 0 77.6%

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

      1. +-commutative 77.6%

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

      2. mul-1-neg 77.6%

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

      3. unsub-neg 77.6%

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

      4. unpow2 77.6%

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

      5. associate-/r* 82.5%

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

      6. div-sub 84.9%

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

      7. associate-/l* 84.7%

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

   5. Simplified 84.7%

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

   if 4.30000000000000013e-66 < c < 1.05000000000000008e69 or 9e117 < c

   1. Initial program 47.2%

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

   3. Taylor expanded in c around inf 76.4%

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

      1. mul-1-neg 76.4%

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

      2. unsub-neg 76.4%

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

      3. associate-/l* 84.5%

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

   5. Simplified 84.5%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;c \leq 4.3 \cdot 10^{-66} \lor \neg \left(c \leq 1.05 \cdot 10^{+69}\right) \land c \leq 9 \cdot 10^{+117}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.22 \cdot 10^{-23}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;c \leq 4.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{+71} \lor \neg \left(c \leq 2.8 \cdot 10^{+118}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.22e-23)
   (/ (- b (/ (* a d) c)) c)
   (if (<= c 4.8e-66)
     (/ (- (/ (* b c) d) a) d)
     (if (or (<= c 1.6e+71) (not (<= c 2.8e+118)))
       (/ (- b (* a (/ d c))) c)
       (/ (- (* b (/ c d)) a) d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.22e-23) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (c <= 4.8e-66) {
		tmp = (((b * c) / d) - a) / d;
	} else if ((c <= 1.6e+71) || !(c <= 2.8e+118)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    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 <= (-1.22d-23)) then
        tmp = (b - ((a * d) / c)) / c
    else if (c <= 4.8d-66) then
        tmp = (((b * c) / d) - a) / d
    else if ((c <= 1.6d+71) .or. (.not. (c <= 2.8d+118))) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = ((b * (c / d)) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.22e-23) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (c <= 4.8e-66) {
		tmp = (((b * c) / d) - a) / d;
	} else if ((c <= 1.6e+71) || !(c <= 2.8e+118)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.22e-23:
		tmp = (b - ((a * d) / c)) / c
	elif c <= 4.8e-66:
		tmp = (((b * c) / d) - a) / d
	elif (c <= 1.6e+71) or not (c <= 2.8e+118):
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = ((b * (c / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.22e-23)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (c <= 4.8e-66)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif ((c <= 1.6e+71) || !(c <= 2.8e+118))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.22e-23)
		tmp = (b - ((a * d) / c)) / c;
	elseif (c <= 4.8e-66)
		tmp = (((b * c) / d) - a) / d;
	elseif ((c <= 1.6e+71) || ~((c <= 2.8e+118)))
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = ((b * (c / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.22e-23], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 4.8e-66], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[Or[LessEqual[c, 1.6e+71], N[Not[LessEqual[c, 2.8e+118]], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.22000000000000007e-23

   1. Initial program 50.3%

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

   3. Taylor expanded in c around inf 79.2%

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

      1. mul-1-neg 79.2%

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

      2. unsub-neg 79.2%

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

      3. associate-/l* 78.0%

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

   5. Simplified 78.0%

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

   6. Taylor expanded in a around 0 79.2%

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

   if -1.22000000000000007e-23 < c < 4.80000000000000052e-66

   1. Initial program 69.8%

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

   3. Taylor expanded in d around inf 87.3%

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

      1. +-commutative 87.3%

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

      2. mul-1-neg 87.3%

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

      3. unsub-neg 87.3%

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

   5. Applied egg-rr 87.3%

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

   if 4.80000000000000052e-66 < c < 1.60000000000000012e71 or 2.79999999999999986e118 < c

   1. Initial program 47.2%

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

   3. Taylor expanded in c around inf 76.4%

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

      1. mul-1-neg 76.4%

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

      2. unsub-neg 76.4%

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

      3. associate-/l* 84.5%

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

   5. Simplified 84.5%

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

   if 1.60000000000000012e71 < c < 2.79999999999999986e118

   1. Initial program 50.0%

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

   3. Taylor expanded in c around 0 47.5%

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

      1. +-commutative 47.5%

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

      2. mul-1-neg 47.5%

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

      3. unsub-neg 47.5%

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

      4. unpow2 47.5%

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

      5. associate-/r* 57.5%

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

      6. div-sub 57.5%

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

      7. associate-/l* 77.2%

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

   5. Simplified 77.2%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.22 \cdot 10^{-23}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;c \leq 4.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{+71} \lor \neg \left(c \leq 2.8 \cdot 10^{+118}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{-d}\\ t_1 := \frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{if}\;c \leq -3.5 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -4.9 \cdot 10^{-188}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -4.1 \cdot 10^{-223}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d}}{d}\\ \mathbf{elif}\;c \leq 5.6 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ a (- d))) (t_1 (/ (- b (* a (/ d c))) c)))
   (if (<= c -3.5e-18)
     t_1
     (if (<= c -4.9e-188)
       t_0
       (if (<= c -4.1e-223)
         (/ (* c (/ b d)) d)
         (if (<= c 5.6e-103) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double t_1 = (b - (a * (d / c))) / c;
	double tmp;
	if (c <= -3.5e-18) {
		tmp = t_1;
	} else if (c <= -4.9e-188) {
		tmp = t_0;
	} else if (c <= -4.1e-223) {
		tmp = (c * (b / d)) / d;
	} else if (c <= 5.6e-103) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    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 - (a * (d / c))) / c
    if (c <= (-3.5d-18)) then
        tmp = t_1
    else if (c <= (-4.9d-188)) then
        tmp = t_0
    else if (c <= (-4.1d-223)) then
        tmp = (c * (b / d)) / d
    else if (c <= 5.6d-103) then
        tmp = t_0
    else
        tmp = t_1
    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 - (a * (d / c))) / c;
	double tmp;
	if (c <= -3.5e-18) {
		tmp = t_1;
	} else if (c <= -4.9e-188) {
		tmp = t_0;
	} else if (c <= -4.1e-223) {
		tmp = (c * (b / d)) / d;
	} else if (c <= 5.6e-103) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = a / -d
	t_1 = (b - (a * (d / c))) / c
	tmp = 0
	if c <= -3.5e-18:
		tmp = t_1
	elif c <= -4.9e-188:
		tmp = t_0
	elif c <= -4.1e-223:
		tmp = (c * (b / d)) / d
	elif c <= 5.6e-103:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(a / Float64(-d))
	t_1 = Float64(Float64(b - Float64(a * Float64(d / c))) / c)
	tmp = 0.0
	if (c <= -3.5e-18)
		tmp = t_1;
	elseif (c <= -4.9e-188)
		tmp = t_0;
	elseif (c <= -4.1e-223)
		tmp = Float64(Float64(c * Float64(b / d)) / d);
	elseif (c <= 5.6e-103)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = a / -d;
	t_1 = (b - (a * (d / c))) / c;
	tmp = 0.0;
	if (c <= -3.5e-18)
		tmp = t_1;
	elseif (c <= -4.9e-188)
		tmp = t_0;
	elseif (c <= -4.1e-223)
		tmp = (c * (b / d)) / d;
	elseif (c <= 5.6e-103)
		tmp = t_0;
	else
		tmp = t_1;
	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[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -3.5e-18], t$95$1, If[LessEqual[c, -4.9e-188], t$95$0, If[LessEqual[c, -4.1e-223], N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 5.6e-103], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.4999999999999999e-18 or 5.60000000000000046e-103 < c

   1. Initial program 50.0%

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

   3. Taylor expanded in c around inf 73.5%

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

      1. mul-1-neg 73.5%

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

      2. unsub-neg 73.5%

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

      3. associate-/l* 76.5%

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

   5. Simplified 76.5%

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

   if -3.4999999999999999e-18 < c < -4.90000000000000004e-188 or -4.10000000000000015e-223 < c < 5.60000000000000046e-103

   1. Initial program 66.7%

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

   3. Taylor expanded in c around 0 71.4%

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

      1. associate-*r/ 71.4%

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

      2. neg-mul-1 71.4%

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

   5. Simplified 71.4%

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

   if -4.90000000000000004e-188 < c < -4.10000000000000015e-223

   1. Initial program 99.8%

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

   3. Taylor expanded in d around inf 99.6%

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

   4. Taylor expanded in a around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -4.9 \cdot 10^{-188}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;c \leq -4.1 \cdot 10^{-223}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d}}{d}\\ \mathbf{elif}\;c \leq 5.6 \cdot 10^{-103}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{-d}\\ \mathbf{if}\;c \leq -1.7 \cdot 10^{-18}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;c \leq -4.9 \cdot 10^{-188}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -3.3 \cdot 10^{-223}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d}}{d}\\ \mathbf{elif}\;c \leq 6.6 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ a (- d))))
   (if (<= c -1.7e-18)
     (/ (- b (/ (* a d) c)) c)
     (if (<= c -4.9e-188)
       t_0
       (if (<= c -3.3e-223)
         (/ (* c (/ b d)) d)
         (if (<= c 6.6e-103) t_0 (/ (- b (* a (/ d c))) c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (c <= -1.7e-18) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (c <= -4.9e-188) {
		tmp = t_0;
	} else if (c <= -3.3e-223) {
		tmp = (c * (b / d)) / d;
	} else if (c <= 6.6e-103) {
		tmp = t_0;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / -d
    if (c <= (-1.7d-18)) then
        tmp = (b - ((a * d) / c)) / c
    else if (c <= (-4.9d-188)) then
        tmp = t_0
    else if (c <= (-3.3d-223)) then
        tmp = (c * (b / d)) / d
    else if (c <= 6.6d-103) then
        tmp = t_0
    else
        tmp = (b - (a * (d / c))) / c
    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 tmp;
	if (c <= -1.7e-18) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (c <= -4.9e-188) {
		tmp = t_0;
	} else if (c <= -3.3e-223) {
		tmp = (c * (b / d)) / d;
	} else if (c <= 6.6e-103) {
		tmp = t_0;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = a / -d
	tmp = 0
	if c <= -1.7e-18:
		tmp = (b - ((a * d) / c)) / c
	elif c <= -4.9e-188:
		tmp = t_0
	elif c <= -3.3e-223:
		tmp = (c * (b / d)) / d
	elif c <= 6.6e-103:
		tmp = t_0
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(a / Float64(-d))
	tmp = 0.0
	if (c <= -1.7e-18)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (c <= -4.9e-188)
		tmp = t_0;
	elseif (c <= -3.3e-223)
		tmp = Float64(Float64(c * Float64(b / d)) / d);
	elseif (c <= 6.6e-103)
		tmp = t_0;
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = a / -d;
	tmp = 0.0;
	if (c <= -1.7e-18)
		tmp = (b - ((a * d) / c)) / c;
	elseif (c <= -4.9e-188)
		tmp = t_0;
	elseif (c <= -3.3e-223)
		tmp = (c * (b / d)) / d;
	elseif (c <= 6.6e-103)
		tmp = t_0;
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(a / (-d)), $MachinePrecision]}, If[LessEqual[c, -1.7e-18], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -4.9e-188], t$95$0, If[LessEqual[c, -3.3e-223], N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 6.6e-103], t$95$0, N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.70000000000000001e-18

   1. Initial program 48.8%

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

   3. Taylor expanded in c around inf 80.9%

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

      1. mul-1-neg 80.9%

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

      2. unsub-neg 80.9%

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

      3. associate-/l* 79.5%

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

   5. Simplified 79.5%

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

   6. Taylor expanded in a around 0 80.9%

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

   if -1.70000000000000001e-18 < c < -4.90000000000000004e-188 or -3.29999999999999994e-223 < c < 6.59999999999999979e-103

   1. Initial program 66.7%

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

   3. Taylor expanded in c around 0 71.4%

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

      1. associate-*r/ 71.4%

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

      2. neg-mul-1 71.4%

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

   5. Simplified 71.4%

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

   if -4.90000000000000004e-188 < c < -3.29999999999999994e-223

   1. Initial program 99.8%

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

   3. Taylor expanded in d around inf 99.6%

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

   4. Taylor expanded in a around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   if 6.59999999999999979e-103 < c

   1. Initial program 51.0%

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

   3. Taylor expanded in c around inf 67.4%

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

      1. mul-1-neg 67.4%

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

      2. unsub-neg 67.4%

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

      3. associate-/l* 74.0%

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

   5. Simplified 74.0%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.7 \cdot 10^{-18}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;c \leq -4.9 \cdot 10^{-188}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;c \leq -3.3 \cdot 10^{-223}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d}}{d}\\ \mathbf{elif}\;c \leq 6.6 \cdot 10^{-103}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{-d}\\ \mathbf{if}\;c \leq -1.15 \cdot 10^{+17}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -4.9 \cdot 10^{-188}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -1.26 \cdot 10^{-219}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d}\\ \mathbf{elif}\;c \leq 6000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ a (- d))))
   (if (<= c -1.15e+17)
     (/ b c)
     (if (<= c -4.9e-188)
       t_0
       (if (<= c -1.26e-219)
         (* (/ c d) (/ b d))
         (if (<= c 6000000.0) t_0 (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (c <= -1.15e+17) {
		tmp = b / c;
	} else if (c <= -4.9e-188) {
		tmp = t_0;
	} else if (c <= -1.26e-219) {
		tmp = (c / d) * (b / d);
	} else if (c <= 6000000.0) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / -d
    if (c <= (-1.15d+17)) then
        tmp = b / c
    else if (c <= (-4.9d-188)) then
        tmp = t_0
    else if (c <= (-1.26d-219)) then
        tmp = (c / d) * (b / d)
    else if (c <= 6000000.0d0) then
        tmp = t_0
    else
        tmp = b / c
    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 tmp;
	if (c <= -1.15e+17) {
		tmp = b / c;
	} else if (c <= -4.9e-188) {
		tmp = t_0;
	} else if (c <= -1.26e-219) {
		tmp = (c / d) * (b / d);
	} else if (c <= 6000000.0) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = a / -d
	tmp = 0
	if c <= -1.15e+17:
		tmp = b / c
	elif c <= -4.9e-188:
		tmp = t_0
	elif c <= -1.26e-219:
		tmp = (c / d) * (b / d)
	elif c <= 6000000.0:
		tmp = t_0
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(a / Float64(-d))
	tmp = 0.0
	if (c <= -1.15e+17)
		tmp = Float64(b / c);
	elseif (c <= -4.9e-188)
		tmp = t_0;
	elseif (c <= -1.26e-219)
		tmp = Float64(Float64(c / d) * Float64(b / d));
	elseif (c <= 6000000.0)
		tmp = t_0;
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = a / -d;
	tmp = 0.0;
	if (c <= -1.15e+17)
		tmp = b / c;
	elseif (c <= -4.9e-188)
		tmp = t_0;
	elseif (c <= -1.26e-219)
		tmp = (c / d) * (b / d);
	elseif (c <= 6000000.0)
		tmp = t_0;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(a / (-d)), $MachinePrecision]}, If[LessEqual[c, -1.15e+17], N[(b / c), $MachinePrecision], If[LessEqual[c, -4.9e-188], t$95$0, If[LessEqual[c, -1.26e-219], N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6000000.0], t$95$0, N[(b / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.15e17 or 6e6 < c

   1. Initial program 43.5%

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

   3. Taylor expanded in c around inf 66.3%

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

   if -1.15e17 < c < -4.90000000000000004e-188 or -1.26000000000000003e-219 < c < 6e6

   1. Initial program 69.8%

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

   3. Taylor expanded in c around 0 65.7%

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

      1. associate-*r/ 65.7%

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

      2. neg-mul-1 65.7%

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

   5. Simplified 65.7%

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

   if -4.90000000000000004e-188 < c < -1.26000000000000003e-219

   1. Initial program 100.0%

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

   3. Taylor expanded in d around inf 99.7%

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

   4. Taylor expanded in a around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

      1. div-inv 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*l* 99.7%

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

      4. div-inv 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{+17}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -4.9 \cdot 10^{-188}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;c \leq -1.26 \cdot 10^{-219}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d}\\ \mathbf{elif}\;c \leq 6000000:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{-d}\\ \mathbf{if}\;c \leq -8.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -4.9 \cdot 10^{-188}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -4.1 \cdot 10^{-223}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d}}{d}\\ \mathbf{elif}\;c \leq 400000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ a (- d))))
   (if (<= c -8.6e+14)
     (/ b c)
     (if (<= c -4.9e-188)
       t_0
       (if (<= c -4.1e-223)
         (/ (* c (/ b d)) d)
         (if (<= c 400000.0) t_0 (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (c <= -8.6e+14) {
		tmp = b / c;
	} else if (c <= -4.9e-188) {
		tmp = t_0;
	} else if (c <= -4.1e-223) {
		tmp = (c * (b / d)) / d;
	} else if (c <= 400000.0) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / -d
    if (c <= (-8.6d+14)) then
        tmp = b / c
    else if (c <= (-4.9d-188)) then
        tmp = t_0
    else if (c <= (-4.1d-223)) then
        tmp = (c * (b / d)) / d
    else if (c <= 400000.0d0) then
        tmp = t_0
    else
        tmp = b / c
    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 tmp;
	if (c <= -8.6e+14) {
		tmp = b / c;
	} else if (c <= -4.9e-188) {
		tmp = t_0;
	} else if (c <= -4.1e-223) {
		tmp = (c * (b / d)) / d;
	} else if (c <= 400000.0) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = a / -d
	tmp = 0
	if c <= -8.6e+14:
		tmp = b / c
	elif c <= -4.9e-188:
		tmp = t_0
	elif c <= -4.1e-223:
		tmp = (c * (b / d)) / d
	elif c <= 400000.0:
		tmp = t_0
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(a / Float64(-d))
	tmp = 0.0
	if (c <= -8.6e+14)
		tmp = Float64(b / c);
	elseif (c <= -4.9e-188)
		tmp = t_0;
	elseif (c <= -4.1e-223)
		tmp = Float64(Float64(c * Float64(b / d)) / d);
	elseif (c <= 400000.0)
		tmp = t_0;
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = a / -d;
	tmp = 0.0;
	if (c <= -8.6e+14)
		tmp = b / c;
	elseif (c <= -4.9e-188)
		tmp = t_0;
	elseif (c <= -4.1e-223)
		tmp = (c * (b / d)) / d;
	elseif (c <= 400000.0)
		tmp = t_0;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(a / (-d)), $MachinePrecision]}, If[LessEqual[c, -8.6e+14], N[(b / c), $MachinePrecision], If[LessEqual[c, -4.9e-188], t$95$0, If[LessEqual[c, -4.1e-223], N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 400000.0], t$95$0, N[(b / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -8.6e14 or 4e5 < c

   1. Initial program 43.5%

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

   3. Taylor expanded in c around inf 66.3%

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

   if -8.6e14 < c < -4.90000000000000004e-188 or -4.10000000000000015e-223 < c < 4e5

   1. Initial program 69.6%

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

   3. Taylor expanded in c around 0 66.1%

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

      1. associate-*r/ 66.1%

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

      2. neg-mul-1 66.1%

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

   5. Simplified 66.1%

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

   if -4.90000000000000004e-188 < c < -4.10000000000000015e-223

   1. Initial program 99.8%

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

   3. Taylor expanded in d around inf 99.6%

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

   4. Taylor expanded in a around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -4.9 \cdot 10^{-188}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;c \leq -4.1 \cdot 10^{-223}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d}}{d}\\ \mathbf{elif}\;c \leq 400000:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.95 \cdot 10^{+18} \lor \neg \left(c \leq 2600000\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.95e+18) (not (<= c 2600000.0))) (/ b c) (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.95e+18) || !(c <= 2600000.0)) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    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 <= (-1.95d+18)) .or. (.not. (c <= 2600000.0d0))) then
        tmp = b / c
    else
        tmp = a / -d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.95e+18) || !(c <= 2600000.0)) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.95e+18) or not (c <= 2600000.0):
		tmp = b / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.95e+18) || !(c <= 2600000.0))
		tmp = Float64(b / c);
	else
		tmp = Float64(a / Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.95e+18) || ~((c <= 2600000.0)))
		tmp = b / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.95e+18], N[Not[LessEqual[c, 2600000.0]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.95e18 or 2.6e6 < c

   1. Initial program 43.5%

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

   3. Taylor expanded in c around inf 66.3%

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

   if -1.95e18 < c < 2.6e6

   1. Initial program 71.1%

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

   3. Taylor expanded in c around 0 63.6%

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

      1. associate-*r/ 63.6%

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

      2. neg-mul-1 63.6%

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

   5. Simplified 63.6%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.95 \cdot 10^{+18} \lor \neg \left(c \leq 2600000\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 9.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{a}{c} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (/ a c))

C

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

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = a / c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_, d_] := N[(a / c), $MachinePrecision]

Derivation

1. Initial program 58.4%

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

   1. *-un-lft-identity 58.4%

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

   2. add-sqr-sqrt 58.4%

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

   3. times-frac 58.3%

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

   4. hypot-define 58.3%

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

   5. fma-neg 58.3%

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

   6. distribute-rgt-neg-in 58.3%

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

   7. hypot-define 74.3%

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

4. Applied egg-rr 74.3%

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

5. Taylor expanded in c around -inf 31.4%

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

   1. associate-*r/ 31.1%

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

   2. +-commutative 31.1%

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

   3. mul-1-neg 31.1%

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

   4. unsub-neg 31.1%

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

   5. *-commutative 31.1%

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

   6. associate-*l/ 31.4%

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

   7. associate-*r/ 30.8%

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

7. Simplified 30.8%

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

8. Taylor expanded in c around 0 6.4%

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

9. Final simplification 6.4%

   \[\leadsto \frac{a}{c} \]
10. Add Preprocessing

Alternative 12: 42.6% accurate, 5.0× 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

real(8) function code(a, b, c, d)
    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 58.4%

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

3. Taylor expanded in c around inf 41.9%

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

4. Final simplification 41.9%

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

Developer target: 99.4% accurate, 0.1× 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

real(8) function code(a, b, c, d)
    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 2024079 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (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))))