Complex division, real part

Percentage Accurate: 61.3% → 82.0%

Time: 7.8s

Alternatives: 9

Speedup: 2.1×

Specification

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	return ((a * c) + (b * 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 = ((a * c) + (b * d)) / ((c * c) + (d * d))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	return ((a * c) + (b * 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 = ((a * c) + (b * d)) / ((c * c) + (d * d))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{+59}:\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{\frac{a}{d}}{d}\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-167}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 8.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\left(-a\right) - \frac{b \cdot d}{c}\right)\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -7e+59)
   (+ (/ b d) (* c (/ (/ a d) d)))
   (if (<= d -1.15e-167)
     (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
     (if (<= d 8.4e-105)
       (* (/ -1.0 c) (- (- a) (/ (* b d) c)))
       (if (<= d 4.2e+27)
         (* (fma a c (* b d)) (/ 1.0 (pow (hypot c d) 2.0)))
         (/ (+ b (/ a (/ d c))) (hypot c d)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -7e+59) {
		tmp = (b / d) + (c * ((a / d) / d));
	} else if (d <= -1.15e-167) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (d <= 8.4e-105) {
		tmp = (-1.0 / c) * (-a - ((b * d) / c));
	} else if (d <= 4.2e+27) {
		tmp = fma(a, c, (b * d)) * (1.0 / pow(hypot(c, d), 2.0));
	} else {
		tmp = (b + (a / (d / c))) / hypot(c, d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -7e+59)
		tmp = Float64(Float64(b / d) + Float64(c * Float64(Float64(a / d) / d)));
	elseif (d <= -1.15e-167)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 8.4e-105)
		tmp = Float64(Float64(-1.0 / c) * Float64(Float64(-a) - Float64(Float64(b * d) / c)));
	elseif (d <= 4.2e+27)
		tmp = Float64(fma(a, c, Float64(b * d)) * Float64(1.0 / (hypot(c, d) ^ 2.0)));
	else
		tmp = Float64(Float64(b + Float64(a / Float64(d / c))) / hypot(c, d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -7e+59], N[(N[(b / d), $MachinePrecision] + N[(c * N[(N[(a / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.15e-167], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8.4e-105], N[(N[(-1.0 / c), $MachinePrecision] * N[((-a) - N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.2e+27], N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -7e59

   1. Initial program 51.2%

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

   2. Taylor expanded in c around 0 86.6%

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

      1. associate-/l* 83.7%

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

      2. associate-/r/ 88.7%

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

   4. Simplified 88.7%

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

      1. *-un-lft-identity 88.7%

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

      2. pow2 88.7%

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

      3. times-frac 92.8%

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

   6. Applied egg-rr 92.8%

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

      1. associate-*l/ 92.8%

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

      2. *-un-lft-identity 92.8%

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

   8. Applied egg-rr 92.8%

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

   if -7e59 < d < -1.1500000000000001e-167

   1. Initial program 84.3%

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

   if -1.1500000000000001e-167 < d < 8.3999999999999999e-105

   1. Initial program 66.6%

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

      1. *-un-lft-identity 66.6%

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

      2. add-sqr-sqrt 66.6%

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

      3. times-frac 66.6%

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

      4. hypot-def 66.7%

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

      5. fma-def 66.7%

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

      6. hypot-def 85.6%

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

   3. Applied egg-rr 85.6%

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

   4. Taylor expanded in c around -inf 55.4%

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

   5. Taylor expanded in c around -inf 92.6%

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

   if 8.3999999999999999e-105 < d < 4.19999999999999989e27

   1. Initial program 87.9%

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

      1. div-inv 88.0%

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

      2. fma-def 88.0%

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

      3. add-sqr-sqrt 88.0%

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

      4. pow2 88.0%

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

      5. hypot-def 88.0%

         \[\leadsto \mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{{\color{blue}{\left(\mathsf{hypot}\left(c, d\right)\right)}}^{2}} \]

   3. Applied egg-rr 88.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}} \]

   if 4.19999999999999989e27 < d

   1. Initial program 53.0%

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

      1. *-un-lft-identity 53.0%

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

      2. add-sqr-sqrt 53.0%

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

      3. times-frac 53.0%

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

      4. hypot-def 53.0%

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

      5. fma-def 53.0%

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

      6. hypot-def 73.0%

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

   3. Applied egg-rr 73.0%

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

      1. associate-*l/ 73.1%

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

      2. *-un-lft-identity 73.1%

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

   5. Applied egg-rr 73.1%

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

   6. Taylor expanded in c around 0 90.1%

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

      1. associate-/l* 91.7%

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

   8. Simplified 91.7%

      \[\leadsto \frac{\color{blue}{b + \frac{a}{\frac{d}{c}}}}{\mathsf{hypot}\left(c, d\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{+59}:\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{\frac{a}{d}}{d}\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-167}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 8.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\left(-a\right) - \frac{b \cdot d}{c}\right)\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 2: 85.6% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) INFINITY)
   (/ (/ (fma a c (* b d)) (hypot c d)) (hypot c d))
   (* (/ c (hypot c d)) (/ a (hypot c d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((a * c) + (b * d)) / ((c * c) + (d * d))) <= ((double) INFINITY)) {
		tmp = (fma(a, c, (b * d)) / hypot(c, d)) / hypot(c, d);
	} else {
		tmp = (c / hypot(c, d)) * (a / hypot(c, d));
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < +inf.0

   1. Initial program 77.7%

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

      1. *-un-lft-identity 77.7%

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

      2. add-sqr-sqrt 77.7%

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

      3. times-frac 77.7%

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

      4. hypot-def 77.7%

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

      5. fma-def 77.7%

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

      6. hypot-def 96.1%

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

   3. Applied egg-rr 96.1%

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

      1. associate-*l/ 96.4%

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

      2. *-un-lft-identity 96.4%

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

   5. Applied egg-rr 96.4%

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

   if +inf.0 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 0.0%

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

   2. Taylor expanded in a around inf 2.0%

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

      1. *-commutative 2.0%

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

   4. Simplified 2.0%

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

      1. add-sqr-sqrt 2.0%

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

      2. hypot-udef 2.0%

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

      3. hypot-udef 2.0%

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

      4. times-frac 52.0%

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

   6. Applied egg-rr 52.0%

      \[\leadsto \color{blue}{\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 3: 82.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -2.35 \cdot 10^{+60}:\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{\frac{a}{d}}{d}\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-167}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\left(-a\right) - \frac{b \cdot d}{c}\right)\\ \mathbf{elif}\;d \leq 5.7 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= d -2.35e+60)
     (+ (/ b d) (* c (/ (/ a d) d)))
     (if (<= d -1.15e-167)
       t_0
       (if (<= d 1.6e-105)
         (* (/ -1.0 c) (- (- a) (/ (* b d) c)))
         (if (<= d 5.7e+35) t_0 (/ (+ b (/ a (/ d c))) (hypot c d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -2.35e+60) {
		tmp = (b / d) + (c * ((a / d) / d));
	} else if (d <= -1.15e-167) {
		tmp = t_0;
	} else if (d <= 1.6e-105) {
		tmp = (-1.0 / c) * (-a - ((b * d) / c));
	} else if (d <= 5.7e+35) {
		tmp = t_0;
	} else {
		tmp = (b + (a / (d / c))) / hypot(c, d);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -2.35e+60) {
		tmp = (b / d) + (c * ((a / d) / d));
	} else if (d <= -1.15e-167) {
		tmp = t_0;
	} else if (d <= 1.6e-105) {
		tmp = (-1.0 / c) * (-a - ((b * d) / c));
	} else if (d <= 5.7e+35) {
		tmp = t_0;
	} else {
		tmp = (b + (a / (d / c))) / Math.hypot(c, d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -2.35e+60:
		tmp = (b / d) + (c * ((a / d) / d))
	elif d <= -1.15e-167:
		tmp = t_0
	elif d <= 1.6e-105:
		tmp = (-1.0 / c) * (-a - ((b * d) / c))
	elif d <= 5.7e+35:
		tmp = t_0
	else:
		tmp = (b + (a / (d / c))) / math.hypot(c, d)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -2.35e+60)
		tmp = Float64(Float64(b / d) + Float64(c * Float64(Float64(a / d) / d)));
	elseif (d <= -1.15e-167)
		tmp = t_0;
	elseif (d <= 1.6e-105)
		tmp = Float64(Float64(-1.0 / c) * Float64(Float64(-a) - Float64(Float64(b * d) / c)));
	elseif (d <= 5.7e+35)
		tmp = t_0;
	else
		tmp = Float64(Float64(b + Float64(a / Float64(d / c))) / hypot(c, d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -2.35e+60)
		tmp = (b / d) + (c * ((a / d) / d));
	elseif (d <= -1.15e-167)
		tmp = t_0;
	elseif (d <= 1.6e-105)
		tmp = (-1.0 / c) * (-a - ((b * d) / c));
	elseif (d <= 5.7e+35)
		tmp = t_0;
	else
		tmp = (b + (a / (d / c))) / hypot(c, d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.35e+60], N[(N[(b / d), $MachinePrecision] + N[(c * N[(N[(a / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.15e-167], t$95$0, If[LessEqual[d, 1.6e-105], N[(N[(-1.0 / c), $MachinePrecision] * N[((-a) - N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.7e+35], t$95$0, N[(N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.3499999999999999e60

   1. Initial program 51.2%

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

   2. Taylor expanded in c around 0 86.6%

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

      1. associate-/l* 83.7%

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

      2. associate-/r/ 88.7%

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

   4. Simplified 88.7%

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

      1. *-un-lft-identity 88.7%

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

      2. pow2 88.7%

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

      3. times-frac 92.8%

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

   6. Applied egg-rr 92.8%

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

      1. associate-*l/ 92.8%

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

      2. *-un-lft-identity 92.8%

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

   8. Applied egg-rr 92.8%

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

   if -2.3499999999999999e60 < d < -1.1500000000000001e-167 or 1.59999999999999991e-105 < d < 5.69999999999999993e35

   1. Initial program 85.7%

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

   if -1.1500000000000001e-167 < d < 1.59999999999999991e-105

   1. Initial program 66.6%

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

      1. *-un-lft-identity 66.6%

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

      2. add-sqr-sqrt 66.6%

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

      3. times-frac 66.6%

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

      4. hypot-def 66.7%

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

      5. fma-def 66.7%

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

      6. hypot-def 85.6%

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

   3. Applied egg-rr 85.6%

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

   4. Taylor expanded in c around -inf 55.4%

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

   5. Taylor expanded in c around -inf 92.6%

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

   if 5.69999999999999993e35 < d

   1. Initial program 52.2%

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

      1. *-un-lft-identity 52.2%

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

      2. add-sqr-sqrt 52.2%

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

      3. times-frac 52.2%

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

      4. hypot-def 52.2%

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

      5. fma-def 52.2%

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

      6. hypot-def 72.5%

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

   3. Applied egg-rr 72.5%

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

      1. associate-*l/ 72.6%

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

      2. *-un-lft-identity 72.6%

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

   5. Applied egg-rr 72.6%

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

   6. Taylor expanded in c around 0 89.9%

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

      1. associate-/l* 91.6%

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

   8. Simplified 91.6%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.35 \cdot 10^{+60}:\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{\frac{a}{d}}{d}\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-167}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\left(-a\right) - \frac{b \cdot d}{c}\right)\\ \mathbf{elif}\;d \leq 5.7 \cdot 10^{+35}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 4: 81.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{d} + c \cdot \frac{\frac{a}{d}}{d}\\ \mathbf{if}\;d \leq -1.65 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-167}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.18 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\left(-a\right) - \frac{b \cdot d}{c}\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (+ (/ b d) (* c (/ (/ a d) d)))))
   (if (<= d -1.65e+52)
     t_1
     (if (<= d -1.15e-167)
       t_0
       (if (<= d 1.18e-105)
         (* (/ -1.0 c) (- (- a) (/ (* b d) c)))
         (if (<= d 7.5e+28) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (b / d) + (c * ((a / d) / d));
	double tmp;
	if (d <= -1.65e+52) {
		tmp = t_1;
	} else if (d <= -1.15e-167) {
		tmp = t_0;
	} else if (d <= 1.18e-105) {
		tmp = (-1.0 / c) * (-a - ((b * d) / c));
	} else if (d <= 7.5e+28) {
		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 * c) + (b * d)) / ((c * c) + (d * d))
    t_1 = (b / d) + (c * ((a / d) / d))
    if (d <= (-1.65d+52)) then
        tmp = t_1
    else if (d <= (-1.15d-167)) then
        tmp = t_0
    else if (d <= 1.18d-105) then
        tmp = ((-1.0d0) / c) * (-a - ((b * d) / c))
    else if (d <= 7.5d+28) 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 * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (b / d) + (c * ((a / d) / d));
	double tmp;
	if (d <= -1.65e+52) {
		tmp = t_1;
	} else if (d <= -1.15e-167) {
		tmp = t_0;
	} else if (d <= 1.18e-105) {
		tmp = (-1.0 / c) * (-a - ((b * d) / c));
	} else if (d <= 7.5e+28) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	t_1 = (b / d) + (c * ((a / d) / d))
	tmp = 0
	if d <= -1.65e+52:
		tmp = t_1
	elif d <= -1.15e-167:
		tmp = t_0
	elif d <= 1.18e-105:
		tmp = (-1.0 / c) * (-a - ((b * d) / c))
	elif d <= 7.5e+28:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b / d) + Float64(c * Float64(Float64(a / d) / d)))
	tmp = 0.0
	if (d <= -1.65e+52)
		tmp = t_1;
	elseif (d <= -1.15e-167)
		tmp = t_0;
	elseif (d <= 1.18e-105)
		tmp = Float64(Float64(-1.0 / c) * Float64(Float64(-a) - Float64(Float64(b * d) / c)));
	elseif (d <= 7.5e+28)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	t_1 = (b / d) + (c * ((a / d) / d));
	tmp = 0.0;
	if (d <= -1.65e+52)
		tmp = t_1;
	elseif (d <= -1.15e-167)
		tmp = t_0;
	elseif (d <= 1.18e-105)
		tmp = (-1.0 / c) * (-a - ((b * d) / c));
	elseif (d <= 7.5e+28)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b / d), $MachinePrecision] + N[(c * N[(N[(a / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.65e+52], t$95$1, If[LessEqual[d, -1.15e-167], t$95$0, If[LessEqual[d, 1.18e-105], N[(N[(-1.0 / c), $MachinePrecision] * N[((-a) - N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.5e+28], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.65e52 or 7.4999999999999998e28 < d

   1. Initial program 52.2%

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

   2. Taylor expanded in c around 0 84.8%

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

      1. associate-/l* 83.6%

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

      2. associate-/r/ 85.8%

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

   4. Simplified 85.8%

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

      1. *-un-lft-identity 85.8%

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

      2. pow2 85.8%

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

      3. times-frac 91.3%

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

   6. Applied egg-rr 91.3%

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

      1. associate-*l/ 91.3%

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

      2. *-un-lft-identity 91.3%

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

   8. Applied egg-rr 91.3%

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

   if -1.65e52 < d < -1.1500000000000001e-167 or 1.1799999999999999e-105 < d < 7.4999999999999998e28

   1. Initial program 85.5%

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

   if -1.1500000000000001e-167 < d < 1.1799999999999999e-105

   1. Initial program 66.6%

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

      1. *-un-lft-identity 66.6%

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

      2. add-sqr-sqrt 66.6%

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

      3. times-frac 66.6%

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

      4. hypot-def 66.7%

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

      5. fma-def 66.7%

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

      6. hypot-def 85.6%

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

   3. Applied egg-rr 85.6%

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

   4. Taylor expanded in c around -inf 55.4%

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

   5. Taylor expanded in c around -inf 92.6%

      \[\leadsto \color{blue}{\frac{-1}{c}} \cdot \left(-1 \cdot a + -1 \cdot \frac{b \cdot d}{c}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.65 \cdot 10^{+52}:\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{\frac{a}{d}}{d}\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-167}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.18 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\left(-a\right) - \frac{b \cdot d}{c}\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+28}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{\frac{a}{d}}{d}\\ \end{array} \]

Alternative 5: 75.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.8 \cdot 10^{-69} \lor \neg \left(d \leq 3.25 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{\frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\left(-a\right) - \frac{b \cdot d}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.8e-69) (not (<= d 3.25e-68)))
   (+ (/ b d) (* c (/ (/ a d) d)))
   (* (/ -1.0 c) (- (- a) (/ (* b d) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.8e-69) || !(d <= 3.25e-68)) {
		tmp = (b / d) + (c * ((a / d) / d));
	} else {
		tmp = (-1.0 / c) * (-a - ((b * d) / 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 ((d <= (-1.8d-69)) .or. (.not. (d <= 3.25d-68))) then
        tmp = (b / d) + (c * ((a / d) / d))
    else
        tmp = ((-1.0d0) / c) * (-a - ((b * d) / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.8e-69) || !(d <= 3.25e-68)) {
		tmp = (b / d) + (c * ((a / d) / d));
	} else {
		tmp = (-1.0 / c) * (-a - ((b * d) / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.8e-69) or not (d <= 3.25e-68):
		tmp = (b / d) + (c * ((a / d) / d))
	else:
		tmp = (-1.0 / c) * (-a - ((b * d) / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.8e-69) || !(d <= 3.25e-68))
		tmp = Float64(Float64(b / d) + Float64(c * Float64(Float64(a / d) / d)));
	else
		tmp = Float64(Float64(-1.0 / c) * Float64(Float64(-a) - Float64(Float64(b * d) / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.8e-69) || ~((d <= 3.25e-68)))
		tmp = (b / d) + (c * ((a / d) / d));
	else
		tmp = (-1.0 / c) * (-a - ((b * d) / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.8e-69], N[Not[LessEqual[d, 3.25e-68]], $MachinePrecision]], N[(N[(b / d), $MachinePrecision] + N[(c * N[(N[(a / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / c), $MachinePrecision] * N[((-a) - N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.80000000000000009e-69 or 3.2499999999999999e-68 < d

   1. Initial program 61.9%

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

   2. Taylor expanded in c around 0 76.5%

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

      1. associate-/l* 75.7%

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

      2. associate-/r/ 77.2%

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

   4. Simplified 77.2%

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

      1. *-un-lft-identity 77.2%

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

      2. pow2 77.2%

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

      3. times-frac 80.8%

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

   6. Applied egg-rr 80.8%

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

      1. associate-*l/ 80.8%

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

      2. *-un-lft-identity 80.8%

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

   8. Applied egg-rr 80.8%

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

   if -1.80000000000000009e-69 < d < 3.2499999999999999e-68

   1. Initial program 73.1%

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

      1. *-un-lft-identity 73.1%

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

      2. add-sqr-sqrt 73.1%

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

      3. times-frac 73.0%

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

      4. hypot-def 73.1%

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

      5. fma-def 73.1%

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

      6. hypot-def 87.7%

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

   3. Applied egg-rr 87.7%

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

   4. Taylor expanded in c around -inf 54.6%

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

   5. Taylor expanded in c around -inf 89.5%

      \[\leadsto \color{blue}{\frac{-1}{c}} \cdot \left(-1 \cdot a + -1 \cdot \frac{b \cdot d}{c}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.8 \cdot 10^{-69} \lor \neg \left(d \leq 3.25 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{\frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\left(-a\right) - \frac{b \cdot d}{c}\right)\\ \end{array} \]

Alternative 6: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{-69} \lor \neg \left(d \leq 1.28 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{\frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.7e-69) (not (<= d 1.28e-70)))
   (+ (/ b d) (* c (/ (/ a d) d)))
   (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.7e-69) || !(d <= 1.28e-70)) {
		tmp = (b / d) + (c * ((a / d) / d));
	} else {
		tmp = a / 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 ((d <= (-1.7d-69)) .or. (.not. (d <= 1.28d-70))) then
        tmp = (b / d) + (c * ((a / d) / d))
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.7e-69) || !(d <= 1.28e-70)) {
		tmp = (b / d) + (c * ((a / d) / d));
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.7e-69) or not (d <= 1.28e-70):
		tmp = (b / d) + (c * ((a / d) / d))
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.7e-69) || !(d <= 1.28e-70))
		tmp = Float64(Float64(b / d) + Float64(c * Float64(Float64(a / d) / d)));
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.7e-69) || ~((d <= 1.28e-70)))
		tmp = (b / d) + (c * ((a / d) / d));
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.7e-69], N[Not[LessEqual[d, 1.28e-70]], $MachinePrecision]], N[(N[(b / d), $MachinePrecision] + N[(c * N[(N[(a / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.70000000000000004e-69 or 1.28e-70 < d

   1. Initial program 61.9%

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

   2. Taylor expanded in c around 0 76.5%

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

      1. associate-/l* 75.7%

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

      2. associate-/r/ 77.2%

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

   4. Simplified 77.2%

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

      1. *-un-lft-identity 77.2%

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

      2. pow2 77.2%

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

      3. times-frac 80.8%

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

   6. Applied egg-rr 80.8%

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

      1. associate-*l/ 80.8%

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

      2. *-un-lft-identity 80.8%

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

   8. Applied egg-rr 80.8%

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

   if -1.70000000000000004e-69 < d < 1.28e-70

   1. Initial program 73.1%

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

   2. Taylor expanded in c around inf 65.1%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{-69} \lor \neg \left(d \leq 1.28 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{\frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 7: 63.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -0.0095 \lor \neg \left(d \leq 1.35 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -0.0095) (not (<= d 1.35e-19))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -0.0095) || !(d <= 1.35e-19)) {
		tmp = b / d;
	} else {
		tmp = a / 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 ((d <= (-0.0095d0)) .or. (.not. (d <= 1.35d-19))) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -0.0095) || !(d <= 1.35e-19)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -0.0095) or not (d <= 1.35e-19):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -0.0095) || !(d <= 1.35e-19))
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -0.0095) || ~((d <= 1.35e-19)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -0.0095], N[Not[LessEqual[d, 1.35e-19]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -0.00949999999999999976 or 1.35e-19 < d

   1. Initial program 58.8%

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

   2. Taylor expanded in c around 0 74.6%

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

   if -0.00949999999999999976 < d < 1.35e-19

   1. Initial program 73.5%

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

   2. Taylor expanded in c around inf 58.1%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -0.0095 \lor \neg \left(d \leq 1.35 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 8: 43.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 4.4 \cdot 10^{+126}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (if (<= d 4.4e+126) (/ a c) (/ a d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 4.4e+126) {
		tmp = a / 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 (d <= 4.4d+126) then
        tmp = a / 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 (d <= 4.4e+126) {
		tmp = a / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= 4.4e+126:
		tmp = a / c
	else:
		tmp = a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= 4.4e+126)
		tmp = Float64(a / c);
	else
		tmp = Float64(a / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= 4.4e+126)
		tmp = a / c;
	else
		tmp = a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, 4.4e+126], N[(a / c), $MachinePrecision], N[(a / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 4.39999999999999997e126

   1. Initial program 70.7%

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

   2. Taylor expanded in c around inf 40.9%

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

   if 4.39999999999999997e126 < d

   1. Initial program 41.7%

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

      1. *-un-lft-identity 41.7%

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

      2. add-sqr-sqrt 41.7%

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

      3. times-frac 41.7%

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

      4. hypot-def 41.7%

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

      5. fma-def 41.7%

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

      6. hypot-def 68.2%

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

   3. Applied egg-rr 68.2%

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

   4. Taylor expanded in c around -inf 3.0%

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

   5. Taylor expanded in d around -inf 3.8%

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

      1. +-commutative 3.8%

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

   7. Simplified 3.8%

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

   8. Taylor expanded in b around 0 19.2%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4.4 \cdot 10^{+126}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]

Alternative 9: 43.0% 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 66.1%

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

2. Taylor expanded in c around inf 35.4%

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

3. Final simplification 35.4%

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

Developer target: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (< (fabs d) (fabs c))
   (/ (+ a (* b (/ d c))) (+ c (* d (/ d c))))
   (/ (+ b (* a (/ 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 = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (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 = (a + (b * (d / c))) / (c + (d * (d / c)))
    else
        tmp = (b + (a * (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 = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (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 = (a + (b * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (b + (a * (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(a + Float64(b * Float64(d / c))) / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(b + Float64(a * 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 = (a + (b * (d / c))) / (c + (d * (d / c)));
	else
		tmp = (b + (a * (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[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d + N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023320 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

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