Complex division, real part

Percentage Accurate: 60.9% → 89.5%

Time: 10.6s

Alternatives: 8

Speedup: 1.1×

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: 60.9% 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: 89.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 2 \cdot 10^{+307}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{a}, c\right)}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e+307)))
     (* (/ (fma b (/ d a) c) (hypot c d)) (/ a (hypot c d)))
     (* (/ 1.0 (hypot c d)) (/ (fma a c (* b d)) (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 ((t_0 <= -((double) INFINITY)) || !(t_0 <= 2e+307)) {
		tmp = (fma(b, (d / a), c) / hypot(c, d)) * (a / hypot(c, d));
	} else {
		tmp = (1.0 / hypot(c, d)) * (fma(a, c, (b * d)) / 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 ((t_0 <= Float64(-Inf)) || !(t_0 <= 2e+307))
		tmp = Float64(Float64(fma(b, Float64(d / a), c) / hypot(c, d)) * Float64(a / hypot(c, d)));
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(a, c, Float64(b * d)) / hypot(c, d)));
	end
	return 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[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 2e+307]], $MachinePrecision]], N[(N[(N[(b * N[(d / a), $MachinePrecision] + c), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / 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 or 1.99999999999999997e307 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 20.5%

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

   3. Taylor expanded in a around inf 20.6%

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

      1. associate-/l* 20.6%

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

   5. Simplified 20.6%

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

      1. *-commutative 20.6%

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

      2. add-sqr-sqrt 20.6%

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

      3. hypot-undefine 20.6%

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

      4. hypot-undefine 20.6%

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

      5. times-frac 76.6%

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

      6. +-commutative 76.6%

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

      7. fma-define 76.6%

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

   7. Applied egg-rr 76.6%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 1.99999999999999997e307

   1. Initial program 83.2%

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

      1. *-un-lft-identity 83.2%

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

      2. associate-*r/ 83.2%

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

      3. fma-define 83.2%

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

      4. add-sqr-sqrt 83.2%

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

      5. times-frac 83.2%

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

      6. fma-define 83.2%

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

      7. hypot-define 83.2%

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

      8. fma-define 83.2%

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

      9. fma-define 83.2%

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

      10. hypot-define 98.9%

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

   4. Applied egg-rr 98.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq -\infty \lor \neg \left(\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+307}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{a}, c\right)}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= t_0 (- INFINITY))
     (/ (+ b (* a (/ c d))) d)
     (if (<= t_0 2e+307)
       (* (/ 1.0 (hypot c d)) (/ (fma a c (* b d)) (hypot c d)))
       (/ (+ b (* c (/ a d))) 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 (t_0 <= -((double) INFINITY)) {
		tmp = (b + (a * (c / d))) / d;
	} else if (t_0 <= 2e+307) {
		tmp = (1.0 / hypot(c, d)) * (fma(a, c, (b * d)) / hypot(c, d));
	} else {
		tmp = (b + (c * (a / d))) / 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 (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (t_0 <= 2e+307)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(a, c, Float64(b * d)) / hypot(c, d)));
	else
		tmp = Float64(Float64(b + Float64(c * Float64(a / d))) / d);
	end
	return 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[t$95$0, (-Infinity)], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[t$95$0, 2e+307], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 44.2%

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

   3. Taylor expanded in d around inf 77.7%

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

      1. associate-/l* 83.5%

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

   5. Simplified 83.5%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 1.99999999999999997e307

   1. Initial program 83.2%

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

      1. *-un-lft-identity 83.2%

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

      2. associate-*r/ 83.2%

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

      3. fma-define 83.2%

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

      4. add-sqr-sqrt 83.2%

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

      5. times-frac 83.2%

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

      6. fma-define 83.2%

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

      7. hypot-define 83.2%

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

      8. fma-define 83.2%

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

      9. fma-define 83.2%

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

      10. hypot-define 98.9%

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

   4. Applied egg-rr 98.9%

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

   if 1.99999999999999997e307 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 15.2%

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

   3. Taylor expanded in d around inf 55.5%

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

      1. associate-/l* 67.3%

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

   5. Simplified 67.3%

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

      1. clear-num 67.3%

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

      2. un-div-inv 67.3%

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

   7. Applied egg-rr 67.3%

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

      1. associate-/r/ 67.3%

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

   9. Simplified 67.3%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq -\infty:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -3 \cdot 10^{-83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-145}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \mathbf{elif}\;c \leq 2.8 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(a + b \cdot \frac{d}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -1.5e-7)
     (* (/ a (hypot c d)) (/ c (hypot c d)))
     (if (<= c -3e-83)
       t_0
       (if (<= c 1.65e-145)
         (/ (+ b (/ a (/ d c))) d)
         (if (<= c 2.8e+46)
           t_0
           (* (/ 1.0 (hypot c d)) (+ a (* b (/ d c))))))))))

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 (c <= -1.5e-7) {
		tmp = (a / hypot(c, d)) * (c / hypot(c, d));
	} else if (c <= -3e-83) {
		tmp = t_0;
	} else if (c <= 1.65e-145) {
		tmp = (b + (a / (d / c))) / d;
	} else if (c <= 2.8e+46) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(c, d)) * (a + (b * (d / c)));
	}
	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 (c <= -1.5e-7) {
		tmp = (a / Math.hypot(c, d)) * (c / Math.hypot(c, d));
	} else if (c <= -3e-83) {
		tmp = t_0;
	} else if (c <= 1.65e-145) {
		tmp = (b + (a / (d / c))) / d;
	} else if (c <= 2.8e+46) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * (a + (b * (d / c)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -1.5e-7:
		tmp = (a / math.hypot(c, d)) * (c / math.hypot(c, d))
	elif c <= -3e-83:
		tmp = t_0
	elif c <= 1.65e-145:
		tmp = (b + (a / (d / c))) / d
	elif c <= 2.8e+46:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(c, d)) * (a + (b * (d / c)))
	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 (c <= -1.5e-7)
		tmp = Float64(Float64(a / hypot(c, d)) * Float64(c / hypot(c, d)));
	elseif (c <= -3e-83)
		tmp = t_0;
	elseif (c <= 1.65e-145)
		tmp = Float64(Float64(b + Float64(a / Float64(d / c))) / d);
	elseif (c <= 2.8e+46)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(a + Float64(b * Float64(d / c))));
	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 (c <= -1.5e-7)
		tmp = (a / hypot(c, d)) * (c / hypot(c, d));
	elseif (c <= -3e-83)
		tmp = t_0;
	elseif (c <= 1.65e-145)
		tmp = (b + (a / (d / c))) / d;
	elseif (c <= 2.8e+46)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(c, d)) * (a + (b * (d / c)));
	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[c, -1.5e-7], N[(N[(a / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -3e-83], t$95$0, If[LessEqual[c, 1.65e-145], N[(N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.8e+46], t$95$0, N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.4999999999999999e-7

   1. Initial program 40.6%

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

   3. Taylor expanded in a around inf 37.7%

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

      1. associate-/l* 34.1%

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

   5. Simplified 34.1%

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

      1. *-commutative 34.1%

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

      2. add-sqr-sqrt 34.1%

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

      3. hypot-undefine 34.1%

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

      4. hypot-undefine 34.1%

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

      5. times-frac 83.5%

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

      6. +-commutative 83.5%

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

      7. fma-define 83.5%

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

   7. Applied egg-rr 83.5%

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

   8. Taylor expanded in b around 0 83.3%

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

   if -1.4999999999999999e-7 < c < -3.0000000000000001e-83 or 1.6499999999999999e-145 < c < 2.80000000000000018e46

   1. Initial program 87.3%

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

   if -3.0000000000000001e-83 < c < 1.6499999999999999e-145

   1. Initial program 65.1%

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

   3. Taylor expanded in d around inf 92.1%

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

      1. associate-/l* 93.1%

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

   5. Simplified 93.1%

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

      1. clear-num 93.1%

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

      2. un-div-inv 93.1%

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

   7. Applied egg-rr 93.1%

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

   if 2.80000000000000018e46 < c

   1. Initial program 41.1%

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

      1. *-un-lft-identity 41.1%

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

      2. associate-*r/ 41.1%

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

      3. fma-define 41.1%

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

      4. add-sqr-sqrt 41.1%

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

      5. times-frac 41.2%

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

      6. fma-define 41.2%

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

      7. hypot-define 41.2%

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

      8. fma-define 41.2%

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

      9. fma-define 41.2%

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

      10. hypot-define 61.3%

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

   4. Applied egg-rr 61.3%

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

   5. Taylor expanded in c around inf 78.9%

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

      1. associate-/l* 83.2%

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

   7. Simplified 83.2%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(a + b \cdot \frac{d}{c}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -3 \cdot 10^{-83}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-145}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \mathbf{elif}\;c \leq 2.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(a + b \cdot \frac{d}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -9.4 \cdot 10^{+64}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{-105}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.25 \cdot 10^{+83}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -9.4e+64)
   (/ (+ b (* a (/ c d))) d)
   (if (<= d -1.75e-105)
     (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
     (if (<= d 2.25e+83)
       (/ (+ a (* b (/ d c))) c)
       (/ (+ b (* c (/ a d))) d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -9.4e+64) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= -1.75e-105) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (d <= 2.25e+83) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b + (c * (a / d))) / 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 <= (-9.4d+64)) then
        tmp = (b + (a * (c / d))) / d
    else if (d <= (-1.75d-105)) then
        tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
    else if (d <= 2.25d+83) then
        tmp = (a + (b * (d / c))) / c
    else
        tmp = (b + (c * (a / d))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -9.4e+64) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= -1.75e-105) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (d <= 2.25e+83) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b + (c * (a / d))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -9.4e+64:
		tmp = (b + (a * (c / d))) / d
	elif d <= -1.75e-105:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	elif d <= 2.25e+83:
		tmp = (a + (b * (d / c))) / c
	else:
		tmp = (b + (c * (a / d))) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -9.4e+64)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (d <= -1.75e-105)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 2.25e+83)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(b + Float64(c * Float64(a / d))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -9.4e+64)
		tmp = (b + (a * (c / d))) / d;
	elseif (d <= -1.75e-105)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	elseif (d <= 2.25e+83)
		tmp = (a + (b * (d / c))) / c;
	else
		tmp = (b + (c * (a / d))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -9.4e+64], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -1.75e-105], 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, 2.25e+83], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -9.40000000000000058e64

   1. Initial program 37.0%

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

   3. Taylor expanded in d around inf 80.0%

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

      1. associate-/l* 86.9%

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

   5. Simplified 86.9%

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

   if -9.40000000000000058e64 < d < -1.75e-105

   1. Initial program 90.7%

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

   if -1.75e-105 < d < 2.25e83

   1. Initial program 68.4%

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

   3. Taylor expanded in c around inf 79.3%

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

      1. associate-/l* 80.9%

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

   5. Simplified 80.9%

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

   if 2.25e83 < d

   1. Initial program 47.5%

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

   3. Taylor expanded in d around inf 73.3%

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

      1. associate-/l* 84.2%

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

   5. Simplified 84.2%

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

      1. clear-num 84.2%

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

      2. un-div-inv 84.3%

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

   7. Applied egg-rr 84.3%

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

      1. associate-/r/ 84.3%

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

   9. Simplified 84.3%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.4 \cdot 10^{+64}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{-105}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.25 \cdot 10^{+83}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.7 \cdot 10^{+37} \lor \neg \left(c \leq 0.053\right):\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.7e+37) (not (<= c 0.053)))
   (/ (+ a (* b (/ d c))) c)
   (/ (+ b (* a (/ c d))) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.7e+37) || !(c <= 0.053)) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b + (a * (c / d))) / 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 <= (-2.7d+37)) .or. (.not. (c <= 0.053d0))) then
        tmp = (a + (b * (d / c))) / c
    else
        tmp = (b + (a * (c / d))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.7e+37) || !(c <= 0.053)) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b + (a * (c / d))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.7e+37) or not (c <= 0.053):
		tmp = (a + (b * (d / c))) / c
	else:
		tmp = (b + (a * (c / d))) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.7e+37) || !(c <= 0.053))
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -2.7e+37) || ~((c <= 0.053)))
		tmp = (a + (b * (d / c))) / c;
	else
		tmp = (b + (a * (c / d))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.7e+37], N[Not[LessEqual[c, 0.053]], $MachinePrecision]], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.69999999999999986e37 or 0.0529999999999999985 < c

   1. Initial program 43.8%

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

   3. Taylor expanded in c around inf 75.0%

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

      1. associate-/l* 78.1%

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

   5. Simplified 78.1%

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

   if -2.69999999999999986e37 < c < 0.0529999999999999985

   1. Initial program 72.7%

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

   3. Taylor expanded in d around inf 82.1%

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

      1. associate-/l* 82.7%

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

   5. Simplified 82.7%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.7 \cdot 10^{+37} \lor \neg \left(c \leq 0.053\right):\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{+76} \lor \neg \left(d \leq 1.15 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.45e+76) (not (<= d 1.15e+84)))
   (/ b d)
   (/ (+ a (* b (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.45e+76) || !(d <= 1.15e+84)) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (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 ((d <= (-1.45d+76)) .or. (.not. (d <= 1.15d+84))) then
        tmp = b / d
    else
        tmp = (a + (b * (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 ((d <= -1.45e+76) || !(d <= 1.15e+84)) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.45e+76) or not (d <= 1.15e+84):
		tmp = b / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.45e+76) || !(d <= 1.15e+84))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.45e+76) || ~((d <= 1.15e+84)))
		tmp = b / d;
	else
		tmp = (a + (b * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.45e+76], N[Not[LessEqual[d, 1.15e+84]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.4500000000000001e76 or 1.1499999999999999e84 < d

   1. Initial program 41.4%

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

   3. Taylor expanded in c around 0 69.5%

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

   if -1.4500000000000001e76 < d < 1.1499999999999999e84

   1. Initial program 72.6%

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

   3. Taylor expanded in c around inf 72.7%

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

      1. associate-/l* 73.9%

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

   5. Simplified 73.9%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{+76} \lor \neg \left(d \leq 1.15 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{-63} \lor \neg \left(d \leq 1.6 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -9.2e-63) (not (<= d 1.6e+86))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -9.2e-63) || !(d <= 1.6e+86)) {
		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 <= (-9.2d-63)) .or. (.not. (d <= 1.6d+86))) 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 <= -9.2e-63) || !(d <= 1.6e+86)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -9.2e-63) or not (d <= 1.6e+86):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -9.2e-63) || !(d <= 1.6e+86))
		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 <= -9.2e-63) || ~((d <= 1.6e+86)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -9.2e-63], N[Not[LessEqual[d, 1.6e+86]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -9.2e-63 or 1.6e86 < d

   1. Initial program 51.4%

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

   3. Taylor expanded in c around 0 63.0%

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

   if -9.2e-63 < d < 1.6e86

   1. Initial program 69.8%

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

   3. Taylor expanded in c around inf 66.3%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{-63} \lor \neg \left(d \leq 1.6 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.5% 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 60.4%

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

3. Taylor expanded in c around inf 43.3%

   \[\leadsto \color{blue}{\frac{a}{c}} \]
4. Add Preprocessing

Developer Target 1: 99.3% 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 2024123 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (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))))