Complex division, real part

Percentage Accurate: 61.5% → 84.8%

Time: 9.4s

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: 61.5% 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: 84.8% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* a c) (* b d))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) INFINITY)
     (/ (/ t_0 (hypot c d)) (hypot c d))
     (* (/ d (hypot c d)) (/ b (hypot c d))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a * c) + (b * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= ((double) INFINITY)) {
		tmp = (t_0 / hypot(c, d)) / hypot(c, d);
	} else {
		tmp = (d / hypot(c, d)) * (b / 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);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 / Math.hypot(c, d)) / Math.hypot(c, d);
	} else {
		tmp = (d / Math.hypot(c, d)) * (b / Math.hypot(c, d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a * c) + (b * d)
	tmp = 0
	if (t_0 / ((c * c) + (d * d))) <= math.inf:
		tmp = (t_0 / math.hypot(c, d)) / math.hypot(c, d)
	else:
		tmp = (d / math.hypot(c, d)) * (b / math.hypot(c, d))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a * c) + Float64(b * d))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= Inf)
		tmp = Float64(Float64(t_0 / hypot(c, d)) / hypot(c, d));
	else
		tmp = Float64(Float64(d / hypot(c, d)) * Float64(b / hypot(c, d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (a * c) + (b * d);
	tmp = 0.0;
	if ((t_0 / ((c * c) + (d * d))) <= Inf)
		tmp = (t_0 / hypot(c, d)) / hypot(c, d);
	else
		tmp = (d / hypot(c, d)) * (b / hypot(c, d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / 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 76.7%

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

      1. +-commutative 76.7%

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

      2. fma-udef 76.8%

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

      3. *-un-lft-identity 76.8%

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

      4. associate-*r/ 76.8%

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

      5. add-sqr-sqrt 76.7%

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

      6. times-frac 76.7%

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

      7. fma-udef 76.7%

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

      8. +-commutative 76.7%

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

      9. hypot-def 76.7%

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

      10. fma-def 76.7%

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

      11. fma-udef 76.7%

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

      12. +-commutative 76.7%

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

      13. hypot-def 93.2%

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

   4. Applied egg-rr 93.2%

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

      1. associate-*l/ 93.4%

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

      2. *-un-lft-identity 93.4%

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

   6. Applied egg-rr 93.4%

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

      1. fma-def 93.4%

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

      2. +-commutative 93.4%

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

   8. Applied egg-rr 93.4%

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

   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. Add Preprocessing

   3. Taylor expanded in a around 0 1.2%

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

      1. *-commutative 1.2%

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

      2. add-sqr-sqrt 1.2%

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

      3. hypot-udef 1.2%

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

      4. hypot-udef 1.2%

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

      5. times-frac 65.2%

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

   5. Applied egg-rr 65.2%

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

4. Final simplification 88.3%

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

Alternative 2: 80.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.4 \cdot 10^{+130}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{1}{\frac{\frac{d}{c}}{a}}}{d}\\ \mathbf{elif}\;d \leq -3.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -3.4e+130)
   (+ (/ b d) (/ (/ 1.0 (/ (/ d c) a)) d))
   (if (<= d -3.4e-105)
     (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
     (if (<= d 3.2e+23)
       (* (/ 1.0 c) (+ a (/ b (/ c d))))
       (* (/ d (hypot c d)) (/ b (hypot c d)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.4e+130) {
		tmp = (b / d) + ((1.0 / ((d / c) / a)) / d);
	} else if (d <= -3.4e-105) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (d <= 3.2e+23) {
		tmp = (1.0 / c) * (a + (b / (c / d)));
	} else {
		tmp = (d / hypot(c, d)) * (b / hypot(c, d));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.4e+130) {
		tmp = (b / d) + ((1.0 / ((d / c) / a)) / d);
	} else if (d <= -3.4e-105) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (d <= 3.2e+23) {
		tmp = (1.0 / c) * (a + (b / (c / d)));
	} else {
		tmp = (d / Math.hypot(c, d)) * (b / Math.hypot(c, d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -3.4e+130:
		tmp = (b / d) + ((1.0 / ((d / c) / a)) / d)
	elif d <= -3.4e-105:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	elif d <= 3.2e+23:
		tmp = (1.0 / c) * (a + (b / (c / d)))
	else:
		tmp = (d / math.hypot(c, d)) * (b / math.hypot(c, d))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -3.4e+130)
		tmp = Float64(Float64(b / d) + Float64(Float64(1.0 / Float64(Float64(d / c) / a)) / d));
	elseif (d <= -3.4e-105)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 3.2e+23)
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(b / Float64(c / d))));
	else
		tmp = Float64(Float64(d / hypot(c, d)) * Float64(b / hypot(c, d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -3.4e+130)
		tmp = (b / d) + ((1.0 / ((d / c) / a)) / d);
	elseif (d <= -3.4e-105)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	elseif (d <= 3.2e+23)
		tmp = (1.0 / c) * (a + (b / (c / d)));
	else
		tmp = (d / hypot(c, d)) * (b / hypot(c, d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -3.4e+130], N[(N[(b / d), $MachinePrecision] + N[(N[(1.0 / N[(N[(d / c), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3.4e-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, 3.2e+23], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.4000000000000001e130

   1. Initial program 41.1%

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

   3. Taylor expanded in c around 0 81.6%

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

      1. associate-/l* 77.6%

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

      2. associate-/r/ 82.0%

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

   5. Simplified 82.0%

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

      1. pow2 82.0%

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

      2. associate-*l/ 81.6%

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

      3. associate-/r* 88.4%

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

   7. Applied egg-rr 88.4%

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

      1. clear-num 88.4%

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

      2. inv-pow 88.4%

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

   9. Applied egg-rr 88.4%

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

       1. unpow-1 88.4%

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

       2. *-commutative 88.4%

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

       3. associate-/r* 88.5%

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

   11. Simplified 88.5%

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

   if -3.4000000000000001e130 < d < -3.39999999999999992e-105

   1. Initial program 81.6%

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

   if -3.39999999999999992e-105 < d < 3.2e23

   1. Initial program 70.7%

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

      1. +-commutative 70.7%

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

      2. fma-udef 70.7%

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

      3. *-un-lft-identity 70.7%

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

      4. associate-*r/ 70.7%

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

      5. add-sqr-sqrt 70.7%

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

      6. times-frac 70.7%

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

      7. fma-udef 70.7%

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

      8. +-commutative 70.7%

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

      9. hypot-def 70.7%

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

      10. fma-def 70.7%

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

      11. fma-udef 70.7%

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

      12. +-commutative 70.7%

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

      13. hypot-def 85.8%

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

   4. Applied egg-rr 85.8%

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

   5. Taylor expanded in c around inf 49.3%

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

      1. associate-/l* 50.2%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in c around inf 89.8%

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

   if 3.2e23 < d

   1. Initial program 46.3%

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

   3. Taylor expanded in a around 0 41.7%

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

      1. *-commutative 41.7%

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

      2. add-sqr-sqrt 41.7%

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

      3. hypot-udef 41.7%

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

      4. hypot-udef 41.7%

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

      5. times-frac 81.5%

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

   5. Applied egg-rr 81.5%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.4 \cdot 10^{+130}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{1}{\frac{\frac{d}{c}}{a}}}{d}\\ \mathbf{elif}\;d \leq -3.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.4 \cdot 10^{+130}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{1}{\frac{\frac{d}{c}}{a}}}{d}\\ \mathbf{elif}\;d \leq -9.5 \cdot 10^{-107}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -3.4e+130)
   (+ (/ b d) (/ (/ 1.0 (/ (/ d c) a)) d))
   (if (<= d -9.5e-107)
     (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
     (if (<= d 6.2e+23)
       (* (/ 1.0 c) (+ a (/ b (/ c d))))
       (+ (/ b d) (/ (* c (/ a d)) d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.4e+130) {
		tmp = (b / d) + ((1.0 / ((d / c) / a)) / d);
	} else if (d <= -9.5e-107) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (d <= 6.2e+23) {
		tmp = (1.0 / c) * (a + (b / (c / d)));
	} else {
		tmp = (b / d) + ((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 <= (-3.4d+130)) then
        tmp = (b / d) + ((1.0d0 / ((d / c) / a)) / d)
    else if (d <= (-9.5d-107)) then
        tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
    else if (d <= 6.2d+23) then
        tmp = (1.0d0 / c) * (a + (b / (c / d)))
    else
        tmp = (b / d) + ((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 <= -3.4e+130) {
		tmp = (b / d) + ((1.0 / ((d / c) / a)) / d);
	} else if (d <= -9.5e-107) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (d <= 6.2e+23) {
		tmp = (1.0 / c) * (a + (b / (c / d)));
	} else {
		tmp = (b / d) + ((c * (a / d)) / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -3.4e+130:
		tmp = (b / d) + ((1.0 / ((d / c) / a)) / d)
	elif d <= -9.5e-107:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	elif d <= 6.2e+23:
		tmp = (1.0 / c) * (a + (b / (c / d)))
	else:
		tmp = (b / d) + ((c * (a / d)) / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -3.4e+130)
		tmp = Float64(Float64(b / d) + Float64(Float64(1.0 / Float64(Float64(d / c) / a)) / d));
	elseif (d <= -9.5e-107)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 6.2e+23)
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(b / Float64(c / d))));
	else
		tmp = Float64(Float64(b / d) + Float64(Float64(c * Float64(a / d)) / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -3.4e+130)
		tmp = (b / d) + ((1.0 / ((d / c) / a)) / d);
	elseif (d <= -9.5e-107)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	elseif (d <= 6.2e+23)
		tmp = (1.0 / c) * (a + (b / (c / d)));
	else
		tmp = (b / d) + ((c * (a / d)) / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -3.4e+130], N[(N[(b / d), $MachinePrecision] + N[(N[(1.0 / N[(N[(d / c), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -9.5e-107], 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, 6.2e+23], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / d), $MachinePrecision] + N[(N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.4000000000000001e130

   1. Initial program 41.1%

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

   3. Taylor expanded in c around 0 81.6%

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

      1. associate-/l* 77.6%

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

      2. associate-/r/ 82.0%

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

   5. Simplified 82.0%

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

      1. pow2 82.0%

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

      2. associate-*l/ 81.6%

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

      3. associate-/r* 88.4%

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

   7. Applied egg-rr 88.4%

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

      1. clear-num 88.4%

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

      2. inv-pow 88.4%

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

   9. Applied egg-rr 88.4%

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

       1. unpow-1 88.4%

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

       2. *-commutative 88.4%

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

       3. associate-/r* 88.5%

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

   11. Simplified 88.5%

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

   if -3.4000000000000001e130 < d < -9.4999999999999999e-107

   1. Initial program 81.6%

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

   if -9.4999999999999999e-107 < d < 6.19999999999999941e23

   1. Initial program 70.7%

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

      1. +-commutative 70.7%

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

      2. fma-udef 70.7%

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

      3. *-un-lft-identity 70.7%

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

      4. associate-*r/ 70.7%

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

      5. add-sqr-sqrt 70.7%

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

      6. times-frac 70.7%

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

      7. fma-udef 70.7%

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

      8. +-commutative 70.7%

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

      9. hypot-def 70.7%

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

      10. fma-def 70.7%

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

      11. fma-udef 70.7%

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

      12. +-commutative 70.7%

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

      13. hypot-def 85.8%

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

   4. Applied egg-rr 85.8%

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

   5. Taylor expanded in c around inf 49.3%

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

      1. associate-/l* 50.2%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in c around inf 89.8%

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

   if 6.19999999999999941e23 < d

   1. Initial program 46.3%

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

   3. Taylor expanded in c around 0 72.8%

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

      1. associate-/l* 74.9%

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

      2. associate-/r/ 76.5%

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

   5. Simplified 76.5%

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

      1. pow2 76.5%

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

      2. associate-*l/ 72.8%

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

      3. associate-/r* 74.8%

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

   7. Applied egg-rr 74.8%

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

      1. expm1-log1p-u 72.8%

         \[\leadsto \frac{b}{d} + \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot c}{d}\right)\right)}}{d} \]

      2. expm1-udef 67.6%

         \[\leadsto \frac{b}{d} + \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{a \cdot c}{d}\right)} - 1}}{d} \]

      3. associate-/l* 70.8%

         \[\leadsto \frac{b}{d} + \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{a}{\frac{d}{c}}}\right)} - 1}{d} \]

      4. div-inv 70.8%

         \[\leadsto \frac{b}{d} + \frac{e^{\mathsf{log1p}\left(\color{blue}{a \cdot \frac{1}{\frac{d}{c}}}\right)} - 1}{d} \]

      5. clear-num 70.8%

         \[\leadsto \frac{b}{d} + \frac{e^{\mathsf{log1p}\left(a \cdot \color{blue}{\frac{c}{d}}\right)} - 1}{d} \]

   9. Applied egg-rr 70.8%

      \[\leadsto \frac{b}{d} + \frac{\color{blue}{e^{\mathsf{log1p}\left(a \cdot \frac{c}{d}\right)} - 1}}{d} \]
   10. Step-by-step derivation

       1. expm1-def 76.0%

          \[\leadsto \frac{b}{d} + \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \frac{c}{d}\right)\right)}}{d} \]

       2. expm1-log1p 78.5%

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

       3. associate-*r/ 74.8%

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

       4. *-commutative 74.8%

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

       5. associate-*r/ 78.5%

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

   11. Simplified 78.5%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.4 \cdot 10^{+130}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{1}{\frac{\frac{d}{c}}{a}}}{d}\\ \mathbf{elif}\;d \leq -9.5 \cdot 10^{-107}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.3 \cdot 10^{+38} \lor \neg \left(d \leq 8.6 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -3.3e+38) (not (<= d 8.6e+23)))
   (/ b d)
   (* (/ 1.0 c) (+ a (/ b (/ c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3.3e+38) || !(d <= 8.6e+23)) {
		tmp = b / d;
	} else {
		tmp = (1.0 / c) * (a + (b / (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 ((d <= (-3.3d+38)) .or. (.not. (d <= 8.6d+23))) then
        tmp = b / d
    else
        tmp = (1.0d0 / c) * (a + (b / (c / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3.3e+38) || !(d <= 8.6e+23)) {
		tmp = b / d;
	} else {
		tmp = (1.0 / c) * (a + (b / (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -3.3e+38) or not (d <= 8.6e+23):
		tmp = b / d
	else:
		tmp = (1.0 / c) * (a + (b / (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -3.3e+38) || !(d <= 8.6e+23))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(b / Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -3.3e+38) || ~((d <= 8.6e+23)))
		tmp = b / d;
	else
		tmp = (1.0 / c) * (a + (b / (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -3.3e+38], N[Not[LessEqual[d, 8.6e+23]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -3.2999999999999999e38 or 8.5999999999999997e23 < d

   1. Initial program 49.8%

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

   3. Taylor expanded in c around 0 70.3%

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

   if -3.2999999999999999e38 < d < 8.5999999999999997e23

   1. Initial program 73.6%

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

      1. +-commutative 73.6%

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

      2. fma-udef 73.6%

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

      3. *-un-lft-identity 73.6%

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

      4. associate-*r/ 73.6%

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

      5. add-sqr-sqrt 73.6%

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

      6. times-frac 73.6%

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

      7. fma-udef 73.6%

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

      8. +-commutative 73.6%

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

      9. hypot-def 73.6%

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

      10. fma-def 73.6%

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

      11. fma-udef 73.6%

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

      12. +-commutative 73.6%

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

      13. hypot-def 87.1%

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

   4. Applied egg-rr 87.1%

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

   5. Taylor expanded in c around inf 44.2%

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

      1. associate-/l* 44.9%

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

   7. Simplified 44.9%

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

   8. Taylor expanded in c around inf 82.6%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.3 \cdot 10^{+38} \lor \neg \left(d \leq 8.6 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -0.0115 \lor \neg \left(d \leq 5 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -0.0115) (not (<= d 5e+24)))
   (+ (/ b d) (* (/ c d) (/ a d)))
   (* (/ 1.0 c) (+ a (/ b (/ c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -0.0115) || !(d <= 5e+24)) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = (1.0 / c) * (a + (b / (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 ((d <= (-0.0115d0)) .or. (.not. (d <= 5d+24))) then
        tmp = (b / d) + ((c / d) * (a / d))
    else
        tmp = (1.0d0 / c) * (a + (b / (c / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -0.0115) || !(d <= 5e+24)) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = (1.0 / c) * (a + (b / (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -0.0115) or not (d <= 5e+24):
		tmp = (b / d) + ((c / d) * (a / d))
	else:
		tmp = (1.0 / c) * (a + (b / (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -0.0115) || !(d <= 5e+24))
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	else
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(b / Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -0.0115) || ~((d <= 5e+24)))
		tmp = (b / d) + ((c / d) * (a / d));
	else
		tmp = (1.0 / c) * (a + (b / (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -0.0115], N[Not[LessEqual[d, 5e+24]], $MachinePrecision]], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -0.0115 or 5.00000000000000045e24 < d

   1. Initial program 52.6%

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

   3. Taylor expanded in c around 0 73.5%

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

      1. associate-/l* 72.3%

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

      2. associate-/r/ 75.3%

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

   5. Simplified 75.3%

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

      1. pow2 75.3%

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

      2. associate-*l/ 73.5%

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

      3. associate-/r* 76.6%

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

   7. Applied egg-rr 76.6%

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

      1. associate-/l/ 73.5%

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

      2. *-commutative 73.5%

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

      3. times-frac 78.3%

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

   9. Applied egg-rr 78.3%

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

   if -0.0115 < d < 5.00000000000000045e24

   1. Initial program 72.6%

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

      1. +-commutative 72.6%

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

      2. fma-udef 72.6%

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

      3. *-un-lft-identity 72.6%

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

      4. associate-*r/ 72.6%

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

      5. add-sqr-sqrt 72.6%

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

      6. times-frac 72.6%

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

      7. fma-udef 72.6%

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

      8. +-commutative 72.6%

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

      9. hypot-def 72.6%

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

      10. fma-def 72.6%

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

      11. fma-udef 72.6%

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

      12. +-commutative 72.6%

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

      13. hypot-def 86.9%

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

   4. Applied egg-rr 86.9%

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

   5. Taylor expanded in c around inf 44.8%

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

      1. associate-/l* 45.5%

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

   7. Simplified 45.5%

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

   8. Taylor expanded in c around inf 85.1%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -0.0115 \lor \neg \left(d \leq 5 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8 \cdot 10^{-5}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -8e-5)
   (+ (/ b d) (* (/ c d) (/ a d)))
   (if (<= d 1.25e+24)
     (* (/ 1.0 c) (+ a (/ b (/ c d))))
     (+ (/ b d) (/ (* c (/ a d)) d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -8e-5) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (d <= 1.25e+24) {
		tmp = (1.0 / c) * (a + (b / (c / d)));
	} else {
		tmp = (b / d) + ((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 <= (-8d-5)) then
        tmp = (b / d) + ((c / d) * (a / d))
    else if (d <= 1.25d+24) then
        tmp = (1.0d0 / c) * (a + (b / (c / d)))
    else
        tmp = (b / d) + ((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 <= -8e-5) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (d <= 1.25e+24) {
		tmp = (1.0 / c) * (a + (b / (c / d)));
	} else {
		tmp = (b / d) + ((c * (a / d)) / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -8e-5:
		tmp = (b / d) + ((c / d) * (a / d))
	elif d <= 1.25e+24:
		tmp = (1.0 / c) * (a + (b / (c / d)))
	else:
		tmp = (b / d) + ((c * (a / d)) / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -8e-5)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	elseif (d <= 1.25e+24)
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(b / Float64(c / d))));
	else
		tmp = Float64(Float64(b / d) + Float64(Float64(c * Float64(a / d)) / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -8e-5)
		tmp = (b / d) + ((c / d) * (a / d));
	elseif (d <= 1.25e+24)
		tmp = (1.0 / c) * (a + (b / (c / d)));
	else
		tmp = (b / d) + ((c * (a / d)) / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -8e-5], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.25e+24], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / d), $MachinePrecision] + N[(N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -8.00000000000000065e-5

   1. Initial program 57.7%

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

   3. Taylor expanded in c around 0 74.0%

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

      1. associate-/l* 70.3%

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

      2. associate-/r/ 74.3%

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

   5. Simplified 74.3%

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

      1. pow2 74.3%

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

      2. associate-*l/ 74.0%

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

      3. associate-/r* 78.1%

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

   7. Applied egg-rr 78.1%

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

      1. associate-/l/ 74.0%

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

      2. *-commutative 74.0%

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

      3. times-frac 78.2%

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

   9. Applied egg-rr 78.2%

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

   if -8.00000000000000065e-5 < d < 1.25000000000000011e24

   1. Initial program 72.6%

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

      1. +-commutative 72.6%

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

      2. fma-udef 72.6%

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

      3. *-un-lft-identity 72.6%

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

      4. associate-*r/ 72.6%

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

      5. add-sqr-sqrt 72.6%

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

      6. times-frac 72.6%

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

      7. fma-udef 72.6%

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

      8. +-commutative 72.6%

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

      9. hypot-def 72.6%

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

      10. fma-def 72.6%

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

      11. fma-udef 72.6%

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

      12. +-commutative 72.6%

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

      13. hypot-def 86.9%

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

   4. Applied egg-rr 86.9%

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

   5. Taylor expanded in c around inf 44.8%

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

      1. associate-/l* 45.5%

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

   7. Simplified 45.5%

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

   8. Taylor expanded in c around inf 85.1%

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

   if 1.25000000000000011e24 < d

   1. Initial program 46.3%

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

   3. Taylor expanded in c around 0 72.8%

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

      1. associate-/l* 74.9%

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

      2. associate-/r/ 76.5%

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

   5. Simplified 76.5%

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

      1. pow2 76.5%

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

      2. associate-*l/ 72.8%

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

      3. associate-/r* 74.8%

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

   7. Applied egg-rr 74.8%

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

      1. expm1-log1p-u 72.8%

         \[\leadsto \frac{b}{d} + \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot c}{d}\right)\right)}}{d} \]

      2. expm1-udef 67.6%

         \[\leadsto \frac{b}{d} + \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{a \cdot c}{d}\right)} - 1}}{d} \]

      3. associate-/l* 70.8%

         \[\leadsto \frac{b}{d} + \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{a}{\frac{d}{c}}}\right)} - 1}{d} \]

      4. div-inv 70.8%

         \[\leadsto \frac{b}{d} + \frac{e^{\mathsf{log1p}\left(\color{blue}{a \cdot \frac{1}{\frac{d}{c}}}\right)} - 1}{d} \]

      5. clear-num 70.8%

         \[\leadsto \frac{b}{d} + \frac{e^{\mathsf{log1p}\left(a \cdot \color{blue}{\frac{c}{d}}\right)} - 1}{d} \]

   9. Applied egg-rr 70.8%

      \[\leadsto \frac{b}{d} + \frac{\color{blue}{e^{\mathsf{log1p}\left(a \cdot \frac{c}{d}\right)} - 1}}{d} \]
   10. Step-by-step derivation

       1. expm1-def 76.0%

          \[\leadsto \frac{b}{d} + \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \frac{c}{d}\right)\right)}}{d} \]

       2. expm1-log1p 78.5%

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

       3. associate-*r/ 74.8%

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

       4. *-commutative 74.8%

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

       5. associate-*r/ 78.5%

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

   11. Simplified 78.5%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8 \cdot 10^{-5}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.8 \cdot 10^{+59} \lor \neg \left(c \leq 2.7 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -3.8e+59) (not (<= c 2.7e+18))) (/ a c) (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.8e+59) || !(c <= 2.7e+18)) {
		tmp = a / c;
	} else {
		tmp = b / 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 <= (-3.8d+59)) .or. (.not. (c <= 2.7d+18))) then
        tmp = a / c
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.8e+59) || !(c <= 2.7e+18)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -3.8e+59) or not (c <= 2.7e+18):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -3.8e+59) || !(c <= 2.7e+18))
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -3.8e+59) || ~((c <= 2.7e+18)))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -3.8e+59], N[Not[LessEqual[c, 2.7e+18]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.8000000000000001e59 or 2.7e18 < c

   1. Initial program 43.6%

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

   3. Taylor expanded in c around inf 73.4%

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

   if -3.8000000000000001e59 < c < 2.7e18

   1. Initial program 75.9%

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

   3. Taylor expanded in c around 0 63.9%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.8 \cdot 10^{+59} \lor \neg \left(c \leq 2.7 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

3. Taylor expanded in c around inf 42.2%

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

4. Final simplification 42.2%

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

Developer target: 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 2024027 
(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))))