Complex division, real part

Percentage Accurate: 61.9% → 89.4%

Time: 7.4s

Alternatives: 8

Speedup: 2.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.9% 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: 89.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+270}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\mathsf{hypot}\left(d, c\right)}}{\frac{\mathsf{hypot}\left(d, c\right)}{b}} + a \cdot \frac{1}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) 2e+270)
   (/ (/ (fma a c (* b d)) (hypot c d)) (hypot c d))
   (+ (/ (/ d (hypot d c)) (/ (hypot d c) b)) (* a (/ 1.0 c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((a * c) + (b * d)) / ((c * c) + (d * d))) <= 2e+270) {
		tmp = (fma(a, c, (b * d)) / hypot(c, d)) / hypot(c, d);
	} else {
		tmp = ((d / hypot(d, c)) / (hypot(d, c) / b)) + (a * (1.0 / c));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d))) <= 2e+270)
		tmp = Float64(Float64(fma(a, c, Float64(b * d)) / hypot(c, d)) / hypot(c, d));
	else
		tmp = Float64(Float64(Float64(d / hypot(d, c)) / Float64(hypot(d, c) / b)) + Float64(a * Float64(1.0 / c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+270], N[(N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(1.0 / c), $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))) < 2.0000000000000001e270

   1. Initial program 81.1%

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

      1. *-un-lft-identity 81.1%

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

      2. add-sqr-sqrt 81.1%

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

      3. times-frac 81.1%

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

      4. hypot-def 81.1%

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

      5. fma-def 81.1%

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

      6. hypot-def 97.6%

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

   3. Applied egg-rr 97.6%

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

      1. associate-*l/ 97.8%

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

      2. *-un-lft-identity 97.8%

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

   5. Applied egg-rr 97.8%

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

   if 2.0000000000000001e270 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 14.9%

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

   2. Taylor expanded in a around 0 8.3%

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

      1. associate-/l* 15.0%

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

      2. associate-/r/ 17.7%

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

      3. unpow2 17.7%

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

      4. unpow2 17.7%

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

      5. fma-udef 17.7%

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

      6. associate-/l* 18.9%

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

      7. associate-/r/ 20.4%

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

      8. unpow2 20.4%

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

      9. unpow2 20.4%

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

      10. fma-udef 20.4%

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

   4. Simplified 20.4%

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

      1. *-un-lft-identity 20.4%

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

      2. add-sqr-sqrt 20.4%

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

      3. times-frac 20.4%

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

      4. fma-udef 20.4%

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

      5. hypot-def 20.4%

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

      6. fma-udef 20.4%

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

      7. hypot-def 68.5%

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

   6. Applied egg-rr 68.5%

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

   7. Taylor expanded in c around inf 69.3%

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

      1. associate-*l* 71.8%

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

      2. associate-*l/ 71.9%

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

      3. *-un-lft-identity 71.9%

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

   9. Applied egg-rr 71.9%

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

       1. associate-/l* 71.9%

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

   11. Simplified 71.9%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+270}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\mathsf{hypot}\left(d, c\right)}}{\frac{\mathsf{hypot}\left(d, c\right)}{b}} + a \cdot \frac{1}{c}\\ \end{array} \]

Alternative 2: 82.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{if}\;d \leq -1.1 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{d}{\mathsf{hypot}\left(d, c\right)}}{\frac{\mathsf{hypot}\left(d, c\right)}{b}} + a \cdot \frac{1}{c}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{+123}:\\ \;\;\;\;b \cdot \frac{d}{t_0} + a \cdot \frac{c}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (+ (/ b d) (* (/ c d) (/ a d)))))
   (if (<= d -1.1e+26)
     t_1
     (if (<= d 1.7e-158)
       (+ (/ (/ d (hypot d c)) (/ (hypot d c) b)) (* a (/ 1.0 c)))
       (if (<= d 1.4e+123) (+ (* b (/ d t_0)) (* a (/ c t_0))) t_1)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = (b / d) + ((c / d) * (a / d));
	double tmp;
	if (d <= -1.1e+26) {
		tmp = t_1;
	} else if (d <= 1.7e-158) {
		tmp = ((d / hypot(d, c)) / (hypot(d, c) / b)) + (a * (1.0 / c));
	} else if (d <= 1.4e+123) {
		tmp = (b * (d / t_0)) + (a * (c / t_0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)))
	tmp = 0.0
	if (d <= -1.1e+26)
		tmp = t_1;
	elseif (d <= 1.7e-158)
		tmp = Float64(Float64(Float64(d / hypot(d, c)) / Float64(hypot(d, c) / b)) + Float64(a * Float64(1.0 / c)));
	elseif (d <= 1.4e+123)
		tmp = Float64(Float64(b * Float64(d / t_0)) + Float64(a * Float64(c / t_0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.1e+26], t$95$1, If[LessEqual[d, 1.7e-158], N[(N[(N[(d / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.4e+123], N[(N[(b * N[(d / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(a * N[(c / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.10000000000000004e26 or 1.40000000000000006e123 < d

   1. Initial program 41.7%

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

   2. Taylor expanded in c around 0 81.7%

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

      1. unpow2 81.7%

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

      2. times-frac 87.0%

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

   4. Simplified 87.0%

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

   if -1.10000000000000004e26 < d < 1.7e-158

   1. Initial program 77.3%

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

   2. Taylor expanded in a around 0 70.0%

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

      1. associate-/l* 69.3%

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

      2. associate-/r/ 71.1%

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

      3. unpow2 71.1%

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

      4. unpow2 71.1%

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

      5. fma-udef 71.1%

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

      6. associate-/l* 65.2%

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

      7. associate-/r/ 73.2%

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

      8. unpow2 73.2%

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

      9. unpow2 73.2%

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

      10. fma-udef 73.2%

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

   4. Simplified 73.2%

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

      1. *-un-lft-identity 73.2%

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

      2. add-sqr-sqrt 73.2%

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

      3. times-frac 73.2%

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

      4. fma-udef 73.2%

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

      5. hypot-def 73.2%

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

      6. fma-udef 73.2%

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

      7. hypot-def 76.4%

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

   6. Applied egg-rr 76.4%

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

   7. Taylor expanded in c around inf 87.3%

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

      1. associate-*l* 89.2%

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

      2. associate-*l/ 89.2%

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

      3. *-un-lft-identity 89.2%

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

   9. Applied egg-rr 89.2%

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

       1. associate-/l* 89.1%

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

   11. Simplified 89.1%

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

   if 1.7e-158 < d < 1.40000000000000006e123

   1. Initial program 79.5%

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

   2. Taylor expanded in a around 0 79.5%

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

      1. associate-/l* 73.8%

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

      2. associate-/r/ 84.1%

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

      3. unpow2 84.1%

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

      4. unpow2 84.1%

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

      5. fma-udef 84.1%

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

      6. associate-/l* 81.9%

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

      7. associate-/r/ 84.5%

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

      8. unpow2 84.5%

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

      9. unpow2 84.5%

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

      10. fma-udef 84.5%

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

   4. Simplified 84.5%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{+26}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{d}{\mathsf{hypot}\left(d, c\right)}}{\frac{\mathsf{hypot}\left(d, c\right)}{b}} + a \cdot \frac{1}{c}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{+123}:\\ \;\;\;\;b \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} + a \cdot \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]

Alternative 3: 80.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{d}{\mathsf{hypot}\left(d, c\right)}}{\frac{\mathsf{hypot}\left(d, c\right)}{b}} + a \cdot \frac{1}{c}\\ t_1 := \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{if}\;d \leq -6.5 \cdot 10^{+23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{-93}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{+65}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ (/ d (hypot d c)) (/ (hypot d c) b)) (* a (/ 1.0 c))))
        (t_1 (+ (/ b d) (* (/ c d) (/ a d)))))
   (if (<= d -6.5e+23)
     t_1
     (if (<= d 1.5e-93)
       t_0
       (if (<= d 3.3e+65)
         (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
         (if (<= d 2.6e+100) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((d / hypot(d, c)) / (hypot(d, c) / b)) + (a * (1.0 / c));
	double t_1 = (b / d) + ((c / d) * (a / d));
	double tmp;
	if (d <= -6.5e+23) {
		tmp = t_1;
	} else if (d <= 1.5e-93) {
		tmp = t_0;
	} else if (d <= 3.3e+65) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (d <= 2.6e+100) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((d / Math.hypot(d, c)) / (Math.hypot(d, c) / b)) + (a * (1.0 / c));
	double t_1 = (b / d) + ((c / d) * (a / d));
	double tmp;
	if (d <= -6.5e+23) {
		tmp = t_1;
	} else if (d <= 1.5e-93) {
		tmp = t_0;
	} else if (d <= 3.3e+65) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (d <= 2.6e+100) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((d / math.hypot(d, c)) / (math.hypot(d, c) / b)) + (a * (1.0 / c))
	t_1 = (b / d) + ((c / d) * (a / d))
	tmp = 0
	if d <= -6.5e+23:
		tmp = t_1
	elif d <= 1.5e-93:
		tmp = t_0
	elif d <= 3.3e+65:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	elif d <= 2.6e+100:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(d / hypot(d, c)) / Float64(hypot(d, c) / b)) + Float64(a * Float64(1.0 / c)))
	t_1 = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)))
	tmp = 0.0
	if (d <= -6.5e+23)
		tmp = t_1;
	elseif (d <= 1.5e-93)
		tmp = t_0;
	elseif (d <= 3.3e+65)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 2.6e+100)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((d / hypot(d, c)) / (hypot(d, c) / b)) + (a * (1.0 / c));
	t_1 = (b / d) + ((c / d) * (a / d));
	tmp = 0.0;
	if (d <= -6.5e+23)
		tmp = t_1;
	elseif (d <= 1.5e-93)
		tmp = t_0;
	elseif (d <= 3.3e+65)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	elseif (d <= 2.6e+100)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(d / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.5e+23], t$95$1, If[LessEqual[d, 1.5e-93], t$95$0, If[LessEqual[d, 3.3e+65], 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, 2.6e+100], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.4999999999999996e23 or 2.6000000000000002e100 < d

   1. Initial program 43.1%

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

   2. Taylor expanded in c around 0 81.4%

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

      1. unpow2 81.4%

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

      2. times-frac 86.5%

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

   4. Simplified 86.5%

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

   if -6.4999999999999996e23 < d < 1.5000000000000001e-93 or 3.30000000000000023e65 < d < 2.6000000000000002e100

   1. Initial program 76.3%

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

   2. Taylor expanded in a around 0 70.0%

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

      1. associate-/l* 68.6%

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

      2. associate-/r/ 72.6%

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

      3. unpow2 72.6%

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

      4. unpow2 72.6%

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

      5. fma-udef 72.6%

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

      6. associate-/l* 66.7%

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

      7. associate-/r/ 74.5%

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

      8. unpow2 74.5%

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

      9. unpow2 74.5%

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

      10. fma-udef 74.5%

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

   4. Simplified 74.5%

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

      1. *-un-lft-identity 74.5%

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

      2. add-sqr-sqrt 74.5%

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

      3. times-frac 74.5%

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

      4. fma-udef 74.5%

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

      5. hypot-def 74.5%

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

      6. fma-udef 74.5%

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

      7. hypot-def 77.2%

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

   6. Applied egg-rr 77.2%

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

   7. Taylor expanded in c around inf 87.3%

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

      1. associate-*l* 88.9%

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

      2. associate-*l/ 89.0%

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

      3. *-un-lft-identity 89.0%

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

   9. Applied egg-rr 89.0%

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

       1. associate-/l* 89.0%

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

   11. Simplified 89.0%

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

   if 1.5000000000000001e-93 < d < 3.30000000000000023e65

   1. Initial program 83.6%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{\frac{d}{\mathsf{hypot}\left(d, c\right)}}{\frac{\mathsf{hypot}\left(d, c\right)}{b}} + a \cdot \frac{1}{c}\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{+65}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{d}{\mathsf{hypot}\left(d, c\right)}}{\frac{\mathsf{hypot}\left(d, c\right)}{b}} + a \cdot \frac{1}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]

Alternative 4: 82.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{if}\;d \leq -1.9 \cdot 10^{+94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -2.15 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 10^{-154}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+82}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (+ (/ b d) (* (/ c d) (/ a d)))))
   (if (<= d -1.9e+94)
     t_1
     (if (<= d -2.15e-150)
       t_0
       (if (<= d 1e-154)
         (+ (/ a c) (/ b (* c (/ c d))))
         (if (<= d 1.7e+82) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (b / d) + ((c / d) * (a / d));
	double tmp;
	if (d <= -1.9e+94) {
		tmp = t_1;
	} else if (d <= -2.15e-150) {
		tmp = t_0;
	} else if (d <= 1e-154) {
		tmp = (a / c) + (b / (c * (c / d)));
	} else if (d <= 1.7e+82) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    t_1 = (b / d) + ((c / d) * (a / d))
    if (d <= (-1.9d+94)) then
        tmp = t_1
    else if (d <= (-2.15d-150)) then
        tmp = t_0
    else if (d <= 1d-154) then
        tmp = (a / c) + (b / (c * (c / d)))
    else if (d <= 1.7d+82) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (b / d) + ((c / d) * (a / d));
	double tmp;
	if (d <= -1.9e+94) {
		tmp = t_1;
	} else if (d <= -2.15e-150) {
		tmp = t_0;
	} else if (d <= 1e-154) {
		tmp = (a / c) + (b / (c * (c / d)));
	} else if (d <= 1.7e+82) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	t_1 = (b / d) + ((c / d) * (a / d))
	tmp = 0
	if d <= -1.9e+94:
		tmp = t_1
	elif d <= -2.15e-150:
		tmp = t_0
	elif d <= 1e-154:
		tmp = (a / c) + (b / (c * (c / d)))
	elif d <= 1.7e+82:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)))
	tmp = 0.0
	if (d <= -1.9e+94)
		tmp = t_1;
	elseif (d <= -2.15e-150)
		tmp = t_0;
	elseif (d <= 1e-154)
		tmp = Float64(Float64(a / c) + Float64(b / Float64(c * Float64(c / d))));
	elseif (d <= 1.7e+82)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	t_1 = (b / d) + ((c / d) * (a / d));
	tmp = 0.0;
	if (d <= -1.9e+94)
		tmp = t_1;
	elseif (d <= -2.15e-150)
		tmp = t_0;
	elseif (d <= 1e-154)
		tmp = (a / c) + (b / (c * (c / d)));
	elseif (d <= 1.7e+82)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.9e+94], t$95$1, If[LessEqual[d, -2.15e-150], t$95$0, If[LessEqual[d, 1e-154], N[(N[(a / c), $MachinePrecision] + N[(b / N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.7e+82], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.8999999999999998e94 or 1.69999999999999997e82 < d

   1. Initial program 40.0%

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

   2. Taylor expanded in c around 0 80.0%

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

      1. unpow2 80.0%

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

      2. times-frac 86.2%

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

   4. Simplified 86.2%

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

   if -1.8999999999999998e94 < d < -2.15000000000000002e-150 or 9.9999999999999997e-155 < d < 1.69999999999999997e82

   1. Initial program 84.0%

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

   if -2.15000000000000002e-150 < d < 9.9999999999999997e-155

   1. Initial program 73.4%

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

   2. Taylor expanded in c around inf 78.6%

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

      1. unpow2 78.6%

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

      2. times-frac 94.0%

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

   4. Simplified 94.0%

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

      1. clear-num 94.0%

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

      2. frac-times 94.1%

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

      3. *-un-lft-identity 94.1%

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

   6. Applied egg-rr 94.1%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+94}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq -2.15 \cdot 10^{-150}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 10^{-154}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+82}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]

Alternative 5: 76.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{+22} \lor \neg \left(d \leq -1.05 \cdot 10^{-37}\right) \land \left(d \leq -2 \cdot 10^{-67} \lor \neg \left(d \leq 800000000\right)\right):\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4.8e+22)
         (and (not (<= d -1.05e-37))
              (or (<= d -2e-67) (not (<= d 800000000.0)))))
   (+ (/ b d) (* (/ c d) (/ a d)))
   (+ (/ a c) (* (/ d c) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.8e+22) || (!(d <= -1.05e-37) && ((d <= -2e-67) || !(d <= 800000000.0)))) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-4.8d+22)) .or. (.not. (d <= (-1.05d-37))) .and. (d <= (-2d-67)) .or. (.not. (d <= 800000000.0d0))) then
        tmp = (b / d) + ((c / d) * (a / d))
    else
        tmp = (a / c) + ((d / c) * (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.8e+22) || (!(d <= -1.05e-37) && ((d <= -2e-67) || !(d <= 800000000.0)))) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4.8e+22) or (not (d <= -1.05e-37) and ((d <= -2e-67) or not (d <= 800000000.0))):
		tmp = (b / d) + ((c / d) * (a / d))
	else:
		tmp = (a / c) + ((d / c) * (b / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4.8e+22) || (!(d <= -1.05e-37) && ((d <= -2e-67) || !(d <= 800000000.0))))
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -4.8e+22) || (~((d <= -1.05e-37)) && ((d <= -2e-67) || ~((d <= 800000000.0)))))
		tmp = (b / d) + ((c / d) * (a / d));
	else
		tmp = (a / c) + ((d / c) * (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4.8e+22], And[N[Not[LessEqual[d, -1.05e-37]], $MachinePrecision], Or[LessEqual[d, -2e-67], N[Not[LessEqual[d, 800000000.0]], $MachinePrecision]]]], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.8e22 or -1.05e-37 < d < -1.99999999999999989e-67 or 8e8 < d

   1. Initial program 50.4%

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

   2. Taylor expanded in c around 0 79.4%

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

      1. unpow2 79.4%

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

      2. times-frac 83.4%

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

   4. Simplified 83.4%

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

   if -4.8e22 < d < -1.05e-37 or -1.99999999999999989e-67 < d < 8e8

   1. Initial program 78.7%

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

   2. Taylor expanded in c around inf 73.9%

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

      1. unpow2 73.9%

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

      2. times-frac 83.2%

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

   4. Simplified 83.2%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{+22} \lor \neg \left(d \leq -1.05 \cdot 10^{-37}\right) \land \left(d \leq -2 \cdot 10^{-67} \lor \neg \left(d \leq 800000000\right)\right):\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]

Alternative 6: 71.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{+30} \lor \neg \left(d \leq 41000000000000\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4e+30) (not (<= d 41000000000000.0)))
   (/ b d)
   (+ (/ a c) (* (/ d c) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4e+30) || !(d <= 41000000000000.0)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-4d+30)) .or. (.not. (d <= 41000000000000.0d0))) then
        tmp = b / d
    else
        tmp = (a / c) + ((d / c) * (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4e+30) || !(d <= 41000000000000.0)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4e+30) or not (d <= 41000000000000.0):
		tmp = b / d
	else:
		tmp = (a / c) + ((d / c) * (b / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4e+30) || !(d <= 41000000000000.0))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -4e+30) || ~((d <= 41000000000000.0)))
		tmp = b / d;
	else
		tmp = (a / c) + ((d / c) * (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4e+30], N[Not[LessEqual[d, 41000000000000.0]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.0000000000000001e30 or 4.1e13 < d

   1. Initial program 47.6%

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

   2. Taylor expanded in c around 0 72.0%

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

   if -4.0000000000000001e30 < d < 4.1e13

   1. Initial program 79.8%

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

   2. Taylor expanded in c around inf 71.0%

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

      1. unpow2 71.0%

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

      2. times-frac 79.8%

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

   4. Simplified 79.8%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{+30} \lor \neg \left(d \leq 41000000000000\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]

Alternative 7: 64.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.8 \cdot 10^{+23}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 3100:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.8e+23) (/ b d) (if (<= d 3100.0) (/ a c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.8e+23) {
		tmp = b / d;
	} else if (d <= 3100.0) {
		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 (d <= (-1.8d+23)) then
        tmp = b / d
    else if (d <= 3100.0d0) 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 (d <= -1.8e+23) {
		tmp = b / d;
	} else if (d <= 3100.0) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.8e+23:
		tmp = b / d
	elif d <= 3100.0:
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.8e+23)
		tmp = Float64(b / d);
	elseif (d <= 3100.0)
		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 (d <= -1.8e+23)
		tmp = b / d;
	elseif (d <= 3100.0)
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.8e+23], N[(b / d), $MachinePrecision], If[LessEqual[d, 3100.0], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.7999999999999999e23 or 3100 < d

   1. Initial program 48.4%

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

   2. Taylor expanded in c around 0 70.8%

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

   if -1.7999999999999999e23 < d < 3100

   1. Initial program 79.5%

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

   2. Taylor expanded in c around inf 68.0%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.8 \cdot 10^{+23}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 3100:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

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 64.8%

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

2. Taylor expanded in c around inf 43.6%

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

3. Final simplification 43.6%

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

Developer target: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (abs(d) < abs(c)) then
        tmp = (a + (b * (d / c))) / (c + (d * (d / c)))
    else
        tmp = (b + (a * (c / d))) / (d + (c * (c / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (Math.abs(d) < Math.abs(c)) {
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023208 
(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))))