Complex division, real part

Percentage Accurate: 61.8% → 85.6%

Time: 10.9s

Alternatives: 15

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: 61.8% 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: 85.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+237}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= t_0 (- INFINITY))
     (/ (fma b (/ d c) a) c)
     (if (<= t_0 2e+237)
       (/ (/ (fma a c (* b d)) (hypot c d)) (hypot c d))
       (+ (/ b d) (* (/ c d) (/ a d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(b, (d / c), a) / c;
	} else if (t_0 <= 2e+237) {
		tmp = (fma(a, c, (b * d)) / hypot(c, d)) / hypot(c, d);
	} else {
		tmp = (b / d) + ((c / d) * (a / d));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (t_0 <= 2e+237)
		tmp = Float64(Float64(fma(a, c, Float64(b * d)) / hypot(c, d)) / hypot(c, d));
	else
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$0, 2e+237], 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[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 44.0%

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

   2. Taylor expanded in c around inf 42.9%

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

      1. unpow2 42.9%

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

   4. Simplified 42.9%

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

      1. +-commutative 42.9%

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

      2. associate-/r* 71.6%

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

      3. frac-add 44.0%

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

      4. *-commutative 44.0%

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

   6. Applied egg-rr 44.0%

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

      1. +-commutative 44.0%

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

      2. *-commutative 44.0%

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

      3. distribute-lft-out 44.0%

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

      4. *-commutative 44.0%

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

      5. *-commutative 44.0%

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

      6. associate-*r/ 44.0%

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

   8. Simplified 44.0%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 0.0%

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

      3. times-frac 0.0%

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

      4. pow1 0.0%

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

      5. pow1 0.0%

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

      6. pow-div 0.0%

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

      7. metadata-eval 0.0%

         \[\leadsto e^{\mathsf{log1p}\left({c}^{\color{blue}{0}} \cdot \frac{a + b \cdot \frac{d}{c}}{c}\right)} - 1 \]

      8. metadata-eval 0.0%

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

      9. *-un-lft-identity 0.0%

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

      10. +-commutative 0.0%

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

      11. fma-def 0.0%

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

   10. Applied egg-rr 0.0%

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

       1. expm1-def 0.0%

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

       2. expm1-log1p 83.2%

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

   12. Simplified 83.2%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 1.99999999999999988e237

   1. Initial program 81.6%

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

      1. *-un-lft-identity 81.6%

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

      2. add-sqr-sqrt 81.6%

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

      3. times-frac 81.5%

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

      4. hypot-def 81.6%

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

      5. fma-def 81.6%

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

      6. hypot-def 98.9%

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

   3. Applied egg-rr 98.9%

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

      1. associate-*l/ 99.2%

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

      2. *-un-lft-identity 99.2%

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

   5. Applied egg-rr 99.2%

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

   if 1.99999999999999988e237 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 15.2%

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

   2. Taylor expanded in c around 0 55.5%

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

      1. unpow2 55.5%

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

      2. times-frac 65.3%

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

   4. Simplified 65.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+237}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]

Alternative 2: 84.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.65 \cdot 10^{+89}:\\ \;\;\;\;\left(b + {\left(\frac{\frac{d}{a}}{c}\right)}^{-1}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -7 \cdot 10^{-105}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{+112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \mathsf{fma}\left(-0.5, \frac{b}{d} \cdot \left(c \cdot \frac{c}{d}\right), \frac{c}{\frac{d}{a}}\right)}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.65e+89)
   (* (+ b (pow (/ (/ d a) c) -1.0)) (/ -1.0 (hypot c d)))
   (if (<= d -7e-105)
     (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
     (if (<= d 3.2e-97)
       (/ (fma b (/ d c) a) c)
       (if (<= d 2.9e+112)
         (/ (fma a c (* b d)) (fma c c (* d d)))
         (/
          (+ b (fma -0.5 (* (/ b d) (* c (/ c d))) (/ c (/ d a))))
          (hypot c d)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.65e+89) {
		tmp = (b + pow(((d / a) / c), -1.0)) * (-1.0 / hypot(c, d));
	} else if (d <= -7e-105) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (d <= 3.2e-97) {
		tmp = fma(b, (d / c), a) / c;
	} else if (d <= 2.9e+112) {
		tmp = fma(a, c, (b * d)) / fma(c, c, (d * d));
	} else {
		tmp = (b + fma(-0.5, ((b / d) * (c * (c / d))), (c / (d / a)))) / hypot(c, d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.65e+89)
		tmp = Float64(Float64(b + (Float64(Float64(d / a) / c) ^ -1.0)) * Float64(-1.0 / hypot(c, d)));
	elseif (d <= -7e-105)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 3.2e-97)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (d <= 2.9e+112)
		tmp = Float64(fma(a, c, Float64(b * d)) / fma(c, c, Float64(d * d)));
	else
		tmp = Float64(Float64(b + fma(-0.5, Float64(Float64(b / d) * Float64(c * Float64(c / d))), Float64(c / Float64(d / a)))) / hypot(c, d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.65e+89], N[(N[(b + N[Power[N[(N[(d / a), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -7e-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-97], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.9e+112], N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(-0.5 * N[(N[(b / d), $MachinePrecision] * N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c / N[(d / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.64999999999999987e89

   1. Initial program 27.0%

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

      1. *-un-lft-identity 27.0%

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

      2. add-sqr-sqrt 27.0%

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

      3. times-frac 27.0%

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

      4. hypot-def 27.0%

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

      5. fma-def 27.0%

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

      6. hypot-def 41.8%

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

   3. Applied egg-rr 41.8%

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

   4. Taylor expanded in d around -inf 85.4%

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

      1. mul-1-neg 85.4%

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

      2. unsub-neg 85.4%

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

      3. mul-1-neg 85.4%

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

      4. associate-/l* 90.3%

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

   6. Simplified 90.3%

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

      1. clear-num 90.3%

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

      2. inv-pow 90.3%

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

   8. Applied egg-rr 90.3%

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

   if -1.64999999999999987e89 < d < -7e-105

   1. Initial program 77.1%

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

   if -7e-105 < d < 3.1999999999999998e-97

   1. Initial program 70.7%

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

   2. Taylor expanded in c around inf 81.0%

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

      1. unpow2 81.0%

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

   4. Simplified 81.0%

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

      1. +-commutative 81.0%

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

      2. associate-/r* 90.7%

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

      3. frac-add 66.0%

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

      4. *-commutative 66.0%

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

   6. Applied egg-rr 66.0%

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

      1. +-commutative 66.0%

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

      2. *-commutative 66.0%

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

      3. distribute-lft-out 66.0%

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

      4. *-commutative 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-*r/ 66.1%

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

   8. Simplified 66.1%

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

      1. expm1-log1p-u 44.8%

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

      2. expm1-udef 25.1%

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

      3. times-frac 32.7%

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

      4. pow1 32.7%

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

      5. pow1 32.7%

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

      6. pow-div 32.7%

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

      7. metadata-eval 32.7%

         \[\leadsto e^{\mathsf{log1p}\left({c}^{\color{blue}{0}} \cdot \frac{a + b \cdot \frac{d}{c}}{c}\right)} - 1 \]

      8. metadata-eval 32.7%

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

      9. *-un-lft-identity 32.7%

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

      10. +-commutative 32.7%

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

      11. fma-def 32.7%

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

   10. Applied egg-rr 32.7%

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

       1. expm1-def 63.1%

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

       2. expm1-log1p 91.9%

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

   12. Simplified 91.9%

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

   if 3.1999999999999998e-97 < d < 2.9000000000000002e112

   1. Initial program 80.9%

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

      1. fma-def 80.9%

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

      2. fma-def 80.9%

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

   3. Simplified 80.9%

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

   if 2.9000000000000002e112 < d

   1. Initial program 39.5%

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

      1. *-un-lft-identity 39.5%

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

      2. add-sqr-sqrt 39.5%

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

      3. times-frac 39.5%

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

      4. hypot-def 39.5%

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

      5. fma-def 39.5%

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

      6. hypot-def 65.4%

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

   3. Applied egg-rr 65.4%

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

      1. associate-*l/ 65.5%

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

      2. *-un-lft-identity 65.5%

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

   5. Applied egg-rr 65.5%

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

   6. Taylor expanded in c around 0 63.7%

      \[\leadsto \frac{\color{blue}{b + \left(\frac{c \cdot a}{d} + -0.5 \cdot \frac{{c}^{2} \cdot b}{{d}^{2}}\right)}}{\mathsf{hypot}\left(c, d\right)} \]
   7. Step-by-step derivation

      1. +-commutative 63.7%

         \[\leadsto \frac{b + \color{blue}{\left(-0.5 \cdot \frac{{c}^{2} \cdot b}{{d}^{2}} + \frac{c \cdot a}{d}\right)}}{\mathsf{hypot}\left(c, d\right)} \]

      2. fma-def 63.7%

         \[\leadsto \frac{b + \color{blue}{\mathsf{fma}\left(-0.5, \frac{{c}^{2} \cdot b}{{d}^{2}}, \frac{c \cdot a}{d}\right)}}{\mathsf{hypot}\left(c, d\right)} \]

      3. unpow2 63.7%

         \[\leadsto \frac{b + \mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(c \cdot c\right)} \cdot b}{{d}^{2}}, \frac{c \cdot a}{d}\right)}{\mathsf{hypot}\left(c, d\right)} \]

      4. *-commutative 63.7%

         \[\leadsto \frac{b + \mathsf{fma}\left(-0.5, \frac{\color{blue}{b \cdot \left(c \cdot c\right)}}{{d}^{2}}, \frac{c \cdot a}{d}\right)}{\mathsf{hypot}\left(c, d\right)} \]

      5. unpow2 63.7%

         \[\leadsto \frac{b + \mathsf{fma}\left(-0.5, \frac{b \cdot \left(c \cdot c\right)}{\color{blue}{d \cdot d}}, \frac{c \cdot a}{d}\right)}{\mathsf{hypot}\left(c, d\right)} \]

      6. times-frac 67.9%

         \[\leadsto \frac{b + \mathsf{fma}\left(-0.5, \color{blue}{\frac{b}{d} \cdot \frac{c \cdot c}{d}}, \frac{c \cdot a}{d}\right)}{\mathsf{hypot}\left(c, d\right)} \]

      7. associate-*r/ 78.0%

         \[\leadsto \frac{b + \mathsf{fma}\left(-0.5, \frac{b}{d} \cdot \color{blue}{\left(c \cdot \frac{c}{d}\right)}, \frac{c \cdot a}{d}\right)}{\mathsf{hypot}\left(c, d\right)} \]

      8. associate-/l* 81.9%

         \[\leadsto \frac{b + \mathsf{fma}\left(-0.5, \frac{b}{d} \cdot \left(c \cdot \frac{c}{d}\right), \color{blue}{\frac{c}{\frac{d}{a}}}\right)}{\mathsf{hypot}\left(c, d\right)} \]

   8. Simplified 81.9%

      \[\leadsto \frac{\color{blue}{b + \mathsf{fma}\left(-0.5, \frac{b}{d} \cdot \left(c \cdot \frac{c}{d}\right), \frac{c}{\frac{d}{a}}\right)}}{\mathsf{hypot}\left(c, d\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.65 \cdot 10^{+89}:\\ \;\;\;\;\left(b + {\left(\frac{\frac{d}{a}}{c}\right)}^{-1}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -7 \cdot 10^{-105}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{+112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \mathsf{fma}\left(-0.5, \frac{b}{d} \cdot \left(c \cdot \frac{c}{d}\right), \frac{c}{\frac{d}{a}}\right)}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 3: 84.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{+82}:\\ \;\;\;\;\left(b + \frac{c}{\frac{d}{a}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+111}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{d} \cdot \left(\frac{-c}{\frac{d}{a}} - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -5.2e+82)
   (* (+ b (/ c (/ d a))) (/ -1.0 (hypot c d)))
   (if (<= d -3.5e-105)
     (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
     (if (<= d 5.5e-87)
       (/ (fma b (/ d c) a) c)
       (if (<= d 2e+111)
         (/ (fma a c (* b d)) (fma c c (* d d)))
         (* (/ -1.0 d) (- (/ (- c) (/ d a)) b)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -5.2e+82) {
		tmp = (b + (c / (d / a))) * (-1.0 / hypot(c, d));
	} else if (d <= -3.5e-105) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (d <= 5.5e-87) {
		tmp = fma(b, (d / c), a) / c;
	} else if (d <= 2e+111) {
		tmp = fma(a, c, (b * d)) / fma(c, c, (d * d));
	} else {
		tmp = (-1.0 / d) * ((-c / (d / a)) - b);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -5.2e+82)
		tmp = Float64(Float64(b + Float64(c / Float64(d / a))) * Float64(-1.0 / hypot(c, d)));
	elseif (d <= -3.5e-105)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 5.5e-87)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (d <= 2e+111)
		tmp = Float64(fma(a, c, Float64(b * d)) / fma(c, c, Float64(d * d)));
	else
		tmp = Float64(Float64(-1.0 / d) * Float64(Float64(Float64(-c) / Float64(d / a)) - b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -5.2e+82], N[(N[(b + N[(c / N[(d / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3.5e-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, 5.5e-87], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2e+111], N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / d), $MachinePrecision] * N[(N[((-c) / N[(d / a), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -5.1999999999999997e82

   1. Initial program 30.9%

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

      1. *-un-lft-identity 30.9%

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

      2. add-sqr-sqrt 30.9%

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

      3. times-frac 31.0%

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

      4. hypot-def 31.0%

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

      5. fma-def 31.0%

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

      6. hypot-def 45.0%

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

   3. Applied egg-rr 45.0%

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

   4. Taylor expanded in d around -inf 86.1%

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

      1. mul-1-neg 86.1%

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

      2. unsub-neg 86.1%

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

      3. mul-1-neg 86.1%

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

      4. associate-/l* 90.8%

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

   6. Simplified 90.8%

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

   if -5.1999999999999997e82 < d < -3.5e-105

   1. Initial program 76.0%

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

   if -3.5e-105 < d < 5.5000000000000004e-87

   1. Initial program 70.7%

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

   2. Taylor expanded in c around inf 81.0%

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

      1. unpow2 81.0%

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

   4. Simplified 81.0%

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

      1. +-commutative 81.0%

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

      2. associate-/r* 90.7%

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

      3. frac-add 66.0%

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

      4. *-commutative 66.0%

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

   6. Applied egg-rr 66.0%

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

      1. +-commutative 66.0%

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

      2. *-commutative 66.0%

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

      3. distribute-lft-out 66.0%

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

      4. *-commutative 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-*r/ 66.1%

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

   8. Simplified 66.1%

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

      1. expm1-log1p-u 44.8%

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

      2. expm1-udef 25.1%

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

      3. times-frac 32.7%

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

      4. pow1 32.7%

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

      5. pow1 32.7%

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

      6. pow-div 32.7%

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

      7. metadata-eval 32.7%

         \[\leadsto e^{\mathsf{log1p}\left({c}^{\color{blue}{0}} \cdot \frac{a + b \cdot \frac{d}{c}}{c}\right)} - 1 \]

      8. metadata-eval 32.7%

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

      9. *-un-lft-identity 32.7%

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

      10. +-commutative 32.7%

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

      11. fma-def 32.7%

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

   10. Applied egg-rr 32.7%

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

       1. expm1-def 63.1%

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

       2. expm1-log1p 91.9%

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

   12. Simplified 91.9%

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

   if 5.5000000000000004e-87 < d < 1.99999999999999991e111

   1. Initial program 80.9%

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

      1. fma-def 80.9%

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

      2. fma-def 80.9%

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

   3. Simplified 80.9%

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

   if 1.99999999999999991e111 < d

   1. Initial program 39.5%

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

      1. *-un-lft-identity 39.5%

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

      2. add-sqr-sqrt 39.5%

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

      3. times-frac 39.5%

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

      4. hypot-def 39.5%

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

      5. fma-def 39.5%

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

      6. hypot-def 65.4%

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

   3. Applied egg-rr 65.4%

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

   4. Taylor expanded in d around -inf 22.7%

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

      1. mul-1-neg 22.7%

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

      2. unsub-neg 22.7%

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

      3. mul-1-neg 22.7%

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

      4. associate-/l* 22.8%

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

   6. Simplified 22.8%

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

   7. Taylor expanded in d around -inf 81.3%

      \[\leadsto \color{blue}{\frac{-1}{d}} \cdot \left(\left(-b\right) - \frac{c}{\frac{d}{a}}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{+82}:\\ \;\;\;\;\left(b + \frac{c}{\frac{d}{a}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+111}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{d} \cdot \left(\frac{-c}{\frac{d}{a}} - b\right)\\ \end{array} \]

Alternative 4: 84.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.85 \cdot 10^{+89}:\\ \;\;\;\;\left(b + {\left(\frac{\frac{d}{a}}{c}\right)}^{-1}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -3.3 \cdot 10^{-102}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+111}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{d} \cdot \left(\frac{-c}{\frac{d}{a}} - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2.85e+89)
   (* (+ b (pow (/ (/ d a) c) -1.0)) (/ -1.0 (hypot c d)))
   (if (<= d -3.3e-102)
     (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
     (if (<= d 1.25e-95)
       (/ (fma b (/ d c) a) c)
       (if (<= d 2e+111)
         (/ (fma a c (* b d)) (fma c c (* d d)))
         (* (/ -1.0 d) (- (/ (- c) (/ d a)) b)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.85e+89) {
		tmp = (b + pow(((d / a) / c), -1.0)) * (-1.0 / hypot(c, d));
	} else if (d <= -3.3e-102) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (d <= 1.25e-95) {
		tmp = fma(b, (d / c), a) / c;
	} else if (d <= 2e+111) {
		tmp = fma(a, c, (b * d)) / fma(c, c, (d * d));
	} else {
		tmp = (-1.0 / d) * ((-c / (d / a)) - b);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2.85e+89)
		tmp = Float64(Float64(b + (Float64(Float64(d / a) / c) ^ -1.0)) * Float64(-1.0 / hypot(c, d)));
	elseif (d <= -3.3e-102)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 1.25e-95)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (d <= 2e+111)
		tmp = Float64(fma(a, c, Float64(b * d)) / fma(c, c, Float64(d * d)));
	else
		tmp = Float64(Float64(-1.0 / d) * Float64(Float64(Float64(-c) / Float64(d / a)) - b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2.85e+89], N[(N[(b + N[Power[N[(N[(d / a), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3.3e-102], 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, 1.25e-95], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2e+111], N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / d), $MachinePrecision] * N[(N[((-c) / N[(d / a), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -2.84999999999999975e89

   1. Initial program 27.0%

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

      1. *-un-lft-identity 27.0%

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

      2. add-sqr-sqrt 27.0%

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

      3. times-frac 27.0%

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

      4. hypot-def 27.0%

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

      5. fma-def 27.0%

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

      6. hypot-def 41.8%

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

   3. Applied egg-rr 41.8%

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

   4. Taylor expanded in d around -inf 85.4%

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

      1. mul-1-neg 85.4%

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

      2. unsub-neg 85.4%

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

      3. mul-1-neg 85.4%

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

      4. associate-/l* 90.3%

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

   6. Simplified 90.3%

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

      1. clear-num 90.3%

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

      2. inv-pow 90.3%

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

   8. Applied egg-rr 90.3%

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

   if -2.84999999999999975e89 < d < -3.3e-102

   1. Initial program 77.1%

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

   if -3.3e-102 < d < 1.2499999999999999e-95

   1. Initial program 70.7%

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

   2. Taylor expanded in c around inf 81.0%

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

      1. unpow2 81.0%

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

   4. Simplified 81.0%

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

      1. +-commutative 81.0%

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

      2. associate-/r* 90.7%

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

      3. frac-add 66.0%

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

      4. *-commutative 66.0%

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

   6. Applied egg-rr 66.0%

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

      1. +-commutative 66.0%

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

      2. *-commutative 66.0%

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

      3. distribute-lft-out 66.0%

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

      4. *-commutative 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-*r/ 66.1%

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

   8. Simplified 66.1%

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

      1. expm1-log1p-u 44.8%

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

      2. expm1-udef 25.1%

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

      3. times-frac 32.7%

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

      4. pow1 32.7%

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

      5. pow1 32.7%

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

      6. pow-div 32.7%

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

      7. metadata-eval 32.7%

         \[\leadsto e^{\mathsf{log1p}\left({c}^{\color{blue}{0}} \cdot \frac{a + b \cdot \frac{d}{c}}{c}\right)} - 1 \]

      8. metadata-eval 32.7%

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

      9. *-un-lft-identity 32.7%

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

      10. +-commutative 32.7%

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

      11. fma-def 32.7%

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

   10. Applied egg-rr 32.7%

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

       1. expm1-def 63.1%

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

       2. expm1-log1p 91.9%

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

   12. Simplified 91.9%

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

   if 1.2499999999999999e-95 < d < 1.99999999999999991e111

   1. Initial program 80.9%

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

      1. fma-def 80.9%

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

      2. fma-def 80.9%

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

   3. Simplified 80.9%

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

   if 1.99999999999999991e111 < d

   1. Initial program 39.5%

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

      1. *-un-lft-identity 39.5%

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

      2. add-sqr-sqrt 39.5%

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

      3. times-frac 39.5%

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

      4. hypot-def 39.5%

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

      5. fma-def 39.5%

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

      6. hypot-def 65.4%

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

   3. Applied egg-rr 65.4%

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

   4. Taylor expanded in d around -inf 22.7%

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

      1. mul-1-neg 22.7%

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

      2. unsub-neg 22.7%

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

      3. mul-1-neg 22.7%

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

      4. associate-/l* 22.8%

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

   6. Simplified 22.8%

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

   7. Taylor expanded in d around -inf 81.3%

      \[\leadsto \color{blue}{\frac{-1}{d}} \cdot \left(\left(-b\right) - \frac{c}{\frac{d}{a}}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.85 \cdot 10^{+89}:\\ \;\;\;\;\left(b + {\left(\frac{\frac{d}{a}}{c}\right)}^{-1}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -3.3 \cdot 10^{-102}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+111}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{d} \cdot \left(\frac{-c}{\frac{d}{a}} - b\right)\\ \end{array} \]

Alternative 5: 84.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -4.5 \cdot 10^{+82}:\\ \;\;\;\;\left(b + \frac{c}{\frac{d}{a}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -6.4 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 4 \cdot 10^{+111}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{d} \cdot \left(\frac{-c}{\frac{d}{a}} - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= d -4.5e+82)
     (* (+ b (/ c (/ d a))) (/ -1.0 (hypot c d)))
     (if (<= d -6.4e-105)
       t_0
       (if (<= d 5.5e-97)
         (/ (fma b (/ d c) a) c)
         (if (<= d 4e+111) t_0 (* (/ -1.0 d) (- (/ (- c) (/ d a)) b))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -4.5e+82) {
		tmp = (b + (c / (d / a))) * (-1.0 / hypot(c, d));
	} else if (d <= -6.4e-105) {
		tmp = t_0;
	} else if (d <= 5.5e-97) {
		tmp = fma(b, (d / c), a) / c;
	} else if (d <= 4e+111) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / d) * ((-c / (d / a)) - b);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -4.5e+82)
		tmp = Float64(Float64(b + Float64(c / Float64(d / a))) * Float64(-1.0 / hypot(c, d)));
	elseif (d <= -6.4e-105)
		tmp = t_0;
	elseif (d <= 5.5e-97)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (d <= 4e+111)
		tmp = t_0;
	else
		tmp = Float64(Float64(-1.0 / d) * Float64(Float64(Float64(-c) / Float64(d / a)) - b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4.5e+82], N[(N[(b + N[(c / N[(d / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -6.4e-105], t$95$0, If[LessEqual[d, 5.5e-97], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 4e+111], t$95$0, N[(N[(-1.0 / d), $MachinePrecision] * N[(N[((-c) / N[(d / a), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -4.4999999999999997e82

   1. Initial program 30.9%

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

      1. *-un-lft-identity 30.9%

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

      2. add-sqr-sqrt 30.9%

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

      3. times-frac 31.0%

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

      4. hypot-def 31.0%

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

      5. fma-def 31.0%

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

      6. hypot-def 45.0%

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

   3. Applied egg-rr 45.0%

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

   4. Taylor expanded in d around -inf 86.1%

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

      1. mul-1-neg 86.1%

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

      2. unsub-neg 86.1%

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

      3. mul-1-neg 86.1%

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

      4. associate-/l* 90.8%

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

   6. Simplified 90.8%

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

   if -4.4999999999999997e82 < d < -6.39999999999999962e-105 or 5.49999999999999948e-97 < d < 3.99999999999999983e111

   1. Initial program 78.4%

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

   if -6.39999999999999962e-105 < d < 5.49999999999999948e-97

   1. Initial program 70.7%

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

   2. Taylor expanded in c around inf 81.0%

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

      1. unpow2 81.0%

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

   4. Simplified 81.0%

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

      1. +-commutative 81.0%

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

      2. associate-/r* 90.7%

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

      3. frac-add 66.0%

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

      4. *-commutative 66.0%

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

   6. Applied egg-rr 66.0%

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

      1. +-commutative 66.0%

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

      2. *-commutative 66.0%

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

      3. distribute-lft-out 66.0%

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

      4. *-commutative 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-*r/ 66.1%

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

   8. Simplified 66.1%

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

      1. expm1-log1p-u 44.8%

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

      2. expm1-udef 25.1%

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

      3. times-frac 32.7%

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

      4. pow1 32.7%

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

      5. pow1 32.7%

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

      6. pow-div 32.7%

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

      7. metadata-eval 32.7%

         \[\leadsto e^{\mathsf{log1p}\left({c}^{\color{blue}{0}} \cdot \frac{a + b \cdot \frac{d}{c}}{c}\right)} - 1 \]

      8. metadata-eval 32.7%

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

      9. *-un-lft-identity 32.7%

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

      10. +-commutative 32.7%

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

      11. fma-def 32.7%

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

   10. Applied egg-rr 32.7%

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

       1. expm1-def 63.1%

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

       2. expm1-log1p 91.9%

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

   12. Simplified 91.9%

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

   if 3.99999999999999983e111 < d

   1. Initial program 39.5%

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

      1. *-un-lft-identity 39.5%

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

      2. add-sqr-sqrt 39.5%

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

      3. times-frac 39.5%

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

      4. hypot-def 39.5%

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

      5. fma-def 39.5%

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

      6. hypot-def 65.4%

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

   3. Applied egg-rr 65.4%

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

   4. Taylor expanded in d around -inf 22.7%

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

      1. mul-1-neg 22.7%

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

      2. unsub-neg 22.7%

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

      3. mul-1-neg 22.7%

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

      4. associate-/l* 22.8%

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

   6. Simplified 22.8%

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

   7. Taylor expanded in d around -inf 81.3%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{+82}:\\ \;\;\;\;\left(b + \frac{c}{\frac{d}{a}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -6.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 4 \cdot 10^{+111}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{d} \cdot \left(\frac{-c}{\frac{d}{a}} - b\right)\\ \end{array} \]

Alternative 6: 84.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -7.4 \cdot 10^{+83}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq -4.2 \cdot 10^{-103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3.9 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 8.2 \cdot 10^{+111}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{d} \cdot \left(\frac{-c}{\frac{d}{a}} - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= d -7.4e+83)
     (+ (/ b d) (* (/ c d) (/ a d)))
     (if (<= d -4.2e-103)
       t_0
       (if (<= d 3.9e-97)
         (/ (fma b (/ d c) a) c)
         (if (<= d 8.2e+111) t_0 (* (/ -1.0 d) (- (/ (- c) (/ d a)) b))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -7.4e+83) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (d <= -4.2e-103) {
		tmp = t_0;
	} else if (d <= 3.9e-97) {
		tmp = fma(b, (d / c), a) / c;
	} else if (d <= 8.2e+111) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / d) * ((-c / (d / a)) - b);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -7.4e+83)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	elseif (d <= -4.2e-103)
		tmp = t_0;
	elseif (d <= 3.9e-97)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (d <= 8.2e+111)
		tmp = t_0;
	else
		tmp = Float64(Float64(-1.0 / d) * Float64(Float64(Float64(-c) / Float64(d / a)) - b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -7.4e+83], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.2e-103], t$95$0, If[LessEqual[d, 3.9e-97], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 8.2e+111], t$95$0, N[(N[(-1.0 / d), $MachinePrecision] * N[(N[((-c) / N[(d / a), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7.4000000000000005e83

   1. Initial program 30.9%

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

   2. Taylor expanded in c around 0 81.0%

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

      1. unpow2 81.0%

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

      2. times-frac 90.8%

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

   4. Simplified 90.8%

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

   if -7.4000000000000005e83 < d < -4.20000000000000009e-103 or 3.8999999999999998e-97 < d < 8.19999999999999973e111

   1. Initial program 78.4%

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

   if -4.20000000000000009e-103 < d < 3.8999999999999998e-97

   1. Initial program 70.7%

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

   2. Taylor expanded in c around inf 81.0%

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

      1. unpow2 81.0%

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

   4. Simplified 81.0%

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

      1. +-commutative 81.0%

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

      2. associate-/r* 90.7%

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

      3. frac-add 66.0%

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

      4. *-commutative 66.0%

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

   6. Applied egg-rr 66.0%

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

      1. +-commutative 66.0%

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

      2. *-commutative 66.0%

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

      3. distribute-lft-out 66.0%

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

      4. *-commutative 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-*r/ 66.1%

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

   8. Simplified 66.1%

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

      1. expm1-log1p-u 44.8%

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

      2. expm1-udef 25.1%

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

      3. times-frac 32.7%

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

      4. pow1 32.7%

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

      5. pow1 32.7%

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

      6. pow-div 32.7%

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

      7. metadata-eval 32.7%

         \[\leadsto e^{\mathsf{log1p}\left({c}^{\color{blue}{0}} \cdot \frac{a + b \cdot \frac{d}{c}}{c}\right)} - 1 \]

      8. metadata-eval 32.7%

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

      9. *-un-lft-identity 32.7%

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

      10. +-commutative 32.7%

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

      11. fma-def 32.7%

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

   10. Applied egg-rr 32.7%

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

       1. expm1-def 63.1%

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

       2. expm1-log1p 91.9%

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

   12. Simplified 91.9%

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

   if 8.19999999999999973e111 < d

   1. Initial program 39.5%

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

      1. *-un-lft-identity 39.5%

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

      2. add-sqr-sqrt 39.5%

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

      3. times-frac 39.5%

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

      4. hypot-def 39.5%

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

      5. fma-def 39.5%

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

      6. hypot-def 65.4%

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

   3. Applied egg-rr 65.4%

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

   4. Taylor expanded in d around -inf 22.7%

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

      1. mul-1-neg 22.7%

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

      2. unsub-neg 22.7%

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

      3. mul-1-neg 22.7%

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

      4. associate-/l* 22.8%

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

   6. Simplified 22.8%

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

   7. Taylor expanded in d around -inf 81.3%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.4 \cdot 10^{+83}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq -4.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.9 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 8.2 \cdot 10^{+111}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{d} \cdot \left(\frac{-c}{\frac{d}{a}} - b\right)\\ \end{array} \]

Alternative 7: 83.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{+83}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq -1.8 \cdot 10^{-104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+111}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{d} \cdot \left(\frac{-c}{\frac{d}{a}} - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= d -2.1e+83)
     (+ (/ b d) (* (/ c d) (/ a d)))
     (if (<= d -1.8e-104)
       t_0
       (if (<= d 3.5e-86)
         (+ (/ a c) (/ (/ b (/ c d)) c))
         (if (<= d 2e+111) t_0 (* (/ -1.0 d) (- (/ (- c) (/ d a)) b))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -2.1e+83) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (d <= -1.8e-104) {
		tmp = t_0;
	} else if (d <= 3.5e-86) {
		tmp = (a / c) + ((b / (c / d)) / c);
	} else if (d <= 2e+111) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / d) * ((-c / (d / a)) - b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    if (d <= (-2.1d+83)) then
        tmp = (b / d) + ((c / d) * (a / d))
    else if (d <= (-1.8d-104)) then
        tmp = t_0
    else if (d <= 3.5d-86) then
        tmp = (a / c) + ((b / (c / d)) / c)
    else if (d <= 2d+111) then
        tmp = t_0
    else
        tmp = ((-1.0d0) / d) * ((-c / (d / a)) - b)
    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 tmp;
	if (d <= -2.1e+83) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (d <= -1.8e-104) {
		tmp = t_0;
	} else if (d <= 3.5e-86) {
		tmp = (a / c) + ((b / (c / d)) / c);
	} else if (d <= 2e+111) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / d) * ((-c / (d / a)) - b);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -2.1e+83:
		tmp = (b / d) + ((c / d) * (a / d))
	elif d <= -1.8e-104:
		tmp = t_0
	elif d <= 3.5e-86:
		tmp = (a / c) + ((b / (c / d)) / c)
	elif d <= 2e+111:
		tmp = t_0
	else:
		tmp = (-1.0 / d) * ((-c / (d / a)) - b)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -2.1e+83)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	elseif (d <= -1.8e-104)
		tmp = t_0;
	elseif (d <= 3.5e-86)
		tmp = Float64(Float64(a / c) + Float64(Float64(b / Float64(c / d)) / c));
	elseif (d <= 2e+111)
		tmp = t_0;
	else
		tmp = Float64(Float64(-1.0 / d) * Float64(Float64(Float64(-c) / Float64(d / a)) - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -2.1e+83)
		tmp = (b / d) + ((c / d) * (a / d));
	elseif (d <= -1.8e-104)
		tmp = t_0;
	elseif (d <= 3.5e-86)
		tmp = (a / c) + ((b / (c / d)) / c);
	elseif (d <= 2e+111)
		tmp = t_0;
	else
		tmp = (-1.0 / d) * ((-c / (d / a)) - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.1e+83], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.8e-104], t$95$0, If[LessEqual[d, 3.5e-86], N[(N[(a / c), $MachinePrecision] + N[(N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2e+111], t$95$0, N[(N[(-1.0 / d), $MachinePrecision] * N[(N[((-c) / N[(d / a), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.10000000000000002e83

   1. Initial program 30.9%

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

   2. Taylor expanded in c around 0 81.0%

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

      1. unpow2 81.0%

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

      2. times-frac 90.8%

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

   4. Simplified 90.8%

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

   if -2.10000000000000002e83 < d < -1.7999999999999999e-104 or 3.50000000000000021e-86 < d < 1.99999999999999991e111

   1. Initial program 78.4%

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

   if -1.7999999999999999e-104 < d < 3.50000000000000021e-86

   1. Initial program 70.7%

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

   2. Taylor expanded in c around inf 81.0%

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

      1. unpow2 81.0%

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

   4. Simplified 81.0%

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

      1. clear-num 81.0%

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

      2. inv-pow 81.0%

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

      3. *-commutative 81.0%

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

      4. times-frac 88.6%

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

   6. Applied egg-rr 88.6%

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

      1. unpow-1 88.6%

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

      2. associate-*l/ 89.6%

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

   8. Simplified 89.6%

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

      1. clear-num 89.6%

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

      2. add-cube-cbrt 89.5%

         \[\leadsto \frac{a}{c} + \frac{\color{blue}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}}}{c \cdot \frac{c}{d}} \]

      3. times-frac 90.6%

         \[\leadsto \frac{a}{c} + \color{blue}{\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{c} \cdot \frac{\sqrt[3]{b}}{\frac{c}{d}}} \]

      4. pow2 90.6%

         \[\leadsto \frac{a}{c} + \frac{\color{blue}{{\left(\sqrt[3]{b}\right)}^{2}}}{c} \cdot \frac{\sqrt[3]{b}}{\frac{c}{d}} \]

   10. Applied egg-rr 90.6%

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

       1. associate-*l/ 90.6%

          \[\leadsto \frac{a}{c} + \color{blue}{\frac{{\left(\sqrt[3]{b}\right)}^{2} \cdot \frac{\sqrt[3]{b}}{\frac{c}{d}}}{c}} \]

       2. associate-*r/ 90.6%

          \[\leadsto \frac{a}{c} + \frac{\color{blue}{\frac{{\left(\sqrt[3]{b}\right)}^{2} \cdot \sqrt[3]{b}}{\frac{c}{d}}}}{c} \]

       3. unpow2 90.6%

          \[\leadsto \frac{a}{c} + \frac{\frac{\color{blue}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)} \cdot \sqrt[3]{b}}{\frac{c}{d}}}{c} \]

       4. rem-3cbrt-lft 90.7%

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

   12. Simplified 90.7%

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

   if 1.99999999999999991e111 < d

   1. Initial program 39.5%

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

      1. *-un-lft-identity 39.5%

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

      2. add-sqr-sqrt 39.5%

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

      3. times-frac 39.5%

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

      4. hypot-def 39.5%

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

      5. fma-def 39.5%

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

      6. hypot-def 65.4%

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

   3. Applied egg-rr 65.4%

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

   4. Taylor expanded in d around -inf 22.7%

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

      1. mul-1-neg 22.7%

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

      2. unsub-neg 22.7%

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

      3. mul-1-neg 22.7%

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

      4. associate-/l* 22.8%

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

   6. Simplified 22.8%

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

   7. Taylor expanded in d around -inf 81.3%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{+83}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq -1.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+111}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{d} \cdot \left(\frac{-c}{\frac{d}{a}} - b\right)\\ \end{array} \]

Alternative 8: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{if}\;d \leq -5.4 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.12 \cdot 10^{-60}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 8 \cdot 10^{+84}:\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ b d) (* (/ c d) (/ a d)))))
   (if (<= d -5.4e+41)
     t_0
     (if (<= d 1.12e-60)
       (+ (/ a c) (/ (/ b (/ c d)) c))
       (if (<= d 8e+84) (/ a (+ c (* d (/ d c)))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b / d) + ((c / d) * (a / d));
	double tmp;
	if (d <= -5.4e+41) {
		tmp = t_0;
	} else if (d <= 1.12e-60) {
		tmp = (a / c) + ((b / (c / d)) / c);
	} else if (d <= 8e+84) {
		tmp = a / (c + (d * (d / c)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b / d) + ((c / d) * (a / d))
    if (d <= (-5.4d+41)) then
        tmp = t_0
    else if (d <= 1.12d-60) then
        tmp = (a / c) + ((b / (c / d)) / c)
    else if (d <= 8d+84) then
        tmp = a / (c + (d * (d / c)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b / d) + ((c / d) * (a / d));
	double tmp;
	if (d <= -5.4e+41) {
		tmp = t_0;
	} else if (d <= 1.12e-60) {
		tmp = (a / c) + ((b / (c / d)) / c);
	} else if (d <= 8e+84) {
		tmp = a / (c + (d * (d / c)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b / d) + ((c / d) * (a / d))
	tmp = 0
	if d <= -5.4e+41:
		tmp = t_0
	elif d <= 1.12e-60:
		tmp = (a / c) + ((b / (c / d)) / c)
	elif d <= 8e+84:
		tmp = a / (c + (d * (d / c)))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)))
	tmp = 0.0
	if (d <= -5.4e+41)
		tmp = t_0;
	elseif (d <= 1.12e-60)
		tmp = Float64(Float64(a / c) + Float64(Float64(b / Float64(c / d)) / c));
	elseif (d <= 8e+84)
		tmp = Float64(a / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b / d) + ((c / d) * (a / d));
	tmp = 0.0;
	if (d <= -5.4e+41)
		tmp = t_0;
	elseif (d <= 1.12e-60)
		tmp = (a / c) + ((b / (c / d)) / c);
	elseif (d <= 8e+84)
		tmp = a / (c + (d * (d / c)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.4e+41], t$95$0, If[LessEqual[d, 1.12e-60], N[(N[(a / c), $MachinePrecision] + N[(N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8e+84], N[(a / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.39999999999999999e41 or 8.00000000000000046e84 < d

   1. Initial program 43.2%

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

   2. Taylor expanded in c around 0 76.6%

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

      1. unpow2 76.6%

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

      2. times-frac 84.1%

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

   4. Simplified 84.1%

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

   if -5.39999999999999999e41 < d < 1.12e-60

   1. Initial program 72.6%

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

   2. Taylor expanded in c around inf 73.1%

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

      1. unpow2 73.1%

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

   4. Simplified 73.1%

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

      1. clear-num 73.1%

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

      2. inv-pow 73.1%

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

      3. *-commutative 73.1%

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

      4. times-frac 78.2%

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

   6. Applied egg-rr 78.2%

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

      1. unpow-1 78.2%

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

      2. associate-*l/ 78.9%

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

   8. Simplified 78.9%

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

      1. clear-num 78.9%

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

      2. add-cube-cbrt 78.8%

         \[\leadsto \frac{a}{c} + \frac{\color{blue}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}}}{c \cdot \frac{c}{d}} \]

      3. times-frac 79.5%

         \[\leadsto \frac{a}{c} + \color{blue}{\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{c} \cdot \frac{\sqrt[3]{b}}{\frac{c}{d}}} \]

      4. pow2 79.5%

         \[\leadsto \frac{a}{c} + \frac{\color{blue}{{\left(\sqrt[3]{b}\right)}^{2}}}{c} \cdot \frac{\sqrt[3]{b}}{\frac{c}{d}} \]

   10. Applied egg-rr 79.5%

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

       1. associate-*l/ 79.5%

          \[\leadsto \frac{a}{c} + \color{blue}{\frac{{\left(\sqrt[3]{b}\right)}^{2} \cdot \frac{\sqrt[3]{b}}{\frac{c}{d}}}{c}} \]

       2. associate-*r/ 79.5%

          \[\leadsto \frac{a}{c} + \frac{\color{blue}{\frac{{\left(\sqrt[3]{b}\right)}^{2} \cdot \sqrt[3]{b}}{\frac{c}{d}}}}{c} \]

       3. unpow2 79.5%

          \[\leadsto \frac{a}{c} + \frac{\frac{\color{blue}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)} \cdot \sqrt[3]{b}}{\frac{c}{d}}}{c} \]

       4. rem-3cbrt-lft 79.6%

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

   12. Simplified 79.6%

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

   if 1.12e-60 < d < 8.00000000000000046e84

   1. Initial program 74.5%

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

   2. Taylor expanded in a around inf 46.1%

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

      1. *-commutative 46.1%

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

      2. associate-/l* 46.7%

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

      3. unpow2 46.7%

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

      4. unpow2 46.7%

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

      5. +-commutative 46.7%

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

      6. fma-udef 46.8%

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

   4. Simplified 46.8%

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

   5. Taylor expanded in c around 0 64.4%

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

      1. unpow2 64.4%

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

      2. *-lft-identity 64.4%

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

      3. times-frac 64.4%

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

      4. /-rgt-identity 64.4%

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

   7. Simplified 64.4%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.4 \cdot 10^{+41}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq 1.12 \cdot 10^{-60}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 8 \cdot 10^{+84}:\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]

Alternative 9: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4 \cdot 10^{-13} \lor \neg \left(c \leq 8.5\right):\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{d} \cdot \left(\frac{-c}{\frac{d}{a}} - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -4e-13) (not (<= c 8.5)))
   (+ (/ a c) (* (/ d c) (/ b c)))
   (* (/ -1.0 d) (- (/ (- c) (/ d a)) b))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4e-13) || !(c <= 8.5)) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else {
		tmp = (-1.0 / d) * ((-c / (d / a)) - b);
	}
	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 <= (-4d-13)) .or. (.not. (c <= 8.5d0))) then
        tmp = (a / c) + ((d / c) * (b / c))
    else
        tmp = ((-1.0d0) / d) * ((-c / (d / a)) - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4e-13) || !(c <= 8.5)) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else {
		tmp = (-1.0 / d) * ((-c / (d / a)) - b);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -4e-13) or not (c <= 8.5):
		tmp = (a / c) + ((d / c) * (b / c))
	else:
		tmp = (-1.0 / d) * ((-c / (d / a)) - b)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -4e-13) || !(c <= 8.5))
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	else
		tmp = Float64(Float64(-1.0 / d) * Float64(Float64(Float64(-c) / Float64(d / a)) - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -4e-13) || ~((c <= 8.5)))
		tmp = (a / c) + ((d / c) * (b / c));
	else
		tmp = (-1.0 / d) * ((-c / (d / a)) - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -4e-13], N[Not[LessEqual[c, 8.5]], $MachinePrecision]], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / d), $MachinePrecision] * N[(N[((-c) / N[(d / a), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -4.0000000000000001e-13 or 8.5 < c

   1. Initial program 52.6%

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

   2. Taylor expanded in c around inf 68.4%

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

      1. unpow2 68.4%

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

      2. times-frac 73.4%

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

   4. Simplified 73.4%

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

   if -4.0000000000000001e-13 < c < 8.5

   1. Initial program 70.7%

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

      1. *-un-lft-identity 70.7%

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

      2. add-sqr-sqrt 70.6%

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

      3. times-frac 70.6%

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

      4. hypot-def 70.7%

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

      5. fma-def 70.7%

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

      6. hypot-def 79.1%

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

   3. Applied egg-rr 79.1%

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

   4. Taylor expanded in d around -inf 46.4%

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

      1. mul-1-neg 46.4%

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

      2. unsub-neg 46.4%

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

      3. mul-1-neg 46.4%

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

      4. associate-/l* 45.7%

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

   6. Simplified 45.7%

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

   7. Taylor expanded in d around -inf 82.2%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4 \cdot 10^{-13} \lor \neg \left(c \leq 8.5\right):\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{d} \cdot \left(\frac{-c}{\frac{d}{a}} - b\right)\\ \end{array} \]

Alternative 10: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{+43}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{-61}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{b}{c \cdot c}\\ \mathbf{elif}\;d \leq 1.18 \cdot 10^{+85}:\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.25e+43)
   (/ b d)
   (if (<= d 4.8e-61)
     (+ (/ a c) (* d (/ b (* c c))))
     (if (<= d 1.18e+85) (/ a (+ c (* d (/ d c)))) (/ b d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.25e+43) {
		tmp = b / d;
	} else if (d <= 4.8e-61) {
		tmp = (a / c) + (d * (b / (c * c)));
	} else if (d <= 1.18e+85) {
		tmp = a / (c + (d * (d / 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 <= (-1.25d+43)) then
        tmp = b / d
    else if (d <= 4.8d-61) then
        tmp = (a / c) + (d * (b / (c * c)))
    else if (d <= 1.18d+85) then
        tmp = a / (c + (d * (d / 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 <= -1.25e+43) {
		tmp = b / d;
	} else if (d <= 4.8e-61) {
		tmp = (a / c) + (d * (b / (c * c)));
	} else if (d <= 1.18e+85) {
		tmp = a / (c + (d * (d / c)));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.25e+43:
		tmp = b / d
	elif d <= 4.8e-61:
		tmp = (a / c) + (d * (b / (c * c)))
	elif d <= 1.18e+85:
		tmp = a / (c + (d * (d / c)))
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.25e+43)
		tmp = Float64(b / d);
	elseif (d <= 4.8e-61)
		tmp = Float64(Float64(a / c) + Float64(d * Float64(b / Float64(c * c))));
	elseif (d <= 1.18e+85)
		tmp = Float64(a / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -1.25e+43)
		tmp = b / d;
	elseif (d <= 4.8e-61)
		tmp = (a / c) + (d * (b / (c * c)));
	elseif (d <= 1.18e+85)
		tmp = a / (c + (d * (d / c)));
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.25e+43], N[(b / d), $MachinePrecision], If[LessEqual[d, 4.8e-61], N[(N[(a / c), $MachinePrecision] + N[(d * N[(b / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.18e+85], N[(a / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.2500000000000001e43 or 1.17999999999999997e85 < d

   1. Initial program 43.2%

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

   2. Taylor expanded in c around 0 72.9%

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

   if -1.2500000000000001e43 < d < 4.8000000000000002e-61

   1. Initial program 72.6%

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

   2. Taylor expanded in c around inf 73.1%

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

      1. unpow2 73.1%

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

   4. Simplified 73.1%

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

   5. Taylor expanded in d around 0 73.1%

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

      1. unpow2 73.1%

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

      2. associate-*r/ 69.8%

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

   7. Simplified 69.8%

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

   if 4.8000000000000002e-61 < d < 1.17999999999999997e85

   1. Initial program 74.5%

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

   2. Taylor expanded in a around inf 46.1%

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

      1. *-commutative 46.1%

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

      2. associate-/l* 46.7%

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

      3. unpow2 46.7%

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

      4. unpow2 46.7%

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

      5. +-commutative 46.7%

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

      6. fma-udef 46.8%

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

   4. Simplified 46.8%

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

   5. Taylor expanded in c around 0 64.4%

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

      1. unpow2 64.4%

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

      2. *-lft-identity 64.4%

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

      3. times-frac 64.4%

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

      4. /-rgt-identity 64.4%

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

   7. Simplified 64.4%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{+43}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{-61}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{b}{c \cdot c}\\ \mathbf{elif}\;d \leq 1.18 \cdot 10^{+85}:\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 11: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{+43}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2.35 \cdot 10^{-60}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+85}:\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -3.8e+43)
   (/ b d)
   (if (<= d 2.35e-60)
     (+ (/ a c) (* (/ d c) (/ b c)))
     (if (<= d 4.5e+85) (/ a (+ c (* d (/ d c)))) (/ b d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.8e+43) {
		tmp = b / d;
	} else if (d <= 2.35e-60) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else if (d <= 4.5e+85) {
		tmp = a / (c + (d * (d / 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 <= (-3.8d+43)) then
        tmp = b / d
    else if (d <= 2.35d-60) then
        tmp = (a / c) + ((d / c) * (b / c))
    else if (d <= 4.5d+85) then
        tmp = a / (c + (d * (d / 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 <= -3.8e+43) {
		tmp = b / d;
	} else if (d <= 2.35e-60) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else if (d <= 4.5e+85) {
		tmp = a / (c + (d * (d / c)));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -3.8e+43:
		tmp = b / d
	elif d <= 2.35e-60:
		tmp = (a / c) + ((d / c) * (b / c))
	elif d <= 4.5e+85:
		tmp = a / (c + (d * (d / c)))
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -3.8e+43)
		tmp = Float64(b / d);
	elseif (d <= 2.35e-60)
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	elseif (d <= 4.5e+85)
		tmp = Float64(a / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -3.8e+43)
		tmp = b / d;
	elseif (d <= 2.35e-60)
		tmp = (a / c) + ((d / c) * (b / c));
	elseif (d <= 4.5e+85)
		tmp = a / (c + (d * (d / c)));
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -3.8e+43], N[(b / d), $MachinePrecision], If[LessEqual[d, 2.35e-60], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.5e+85], N[(a / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.80000000000000008e43 or 4.50000000000000007e85 < d

   1. Initial program 43.2%

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

   2. Taylor expanded in c around 0 72.9%

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

   if -3.80000000000000008e43 < d < 2.35e-60

   1. Initial program 72.6%

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

   2. Taylor expanded in c around inf 73.1%

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

      1. unpow2 73.1%

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

      2. times-frac 78.2%

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

   4. Simplified 78.2%

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

   if 2.35e-60 < d < 4.50000000000000007e85

   1. Initial program 74.5%

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

   2. Taylor expanded in a around inf 46.1%

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

      1. *-commutative 46.1%

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

      2. associate-/l* 46.7%

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

      3. unpow2 46.7%

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

      4. unpow2 46.7%

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

      5. +-commutative 46.7%

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

      6. fma-udef 46.8%

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

   4. Simplified 46.8%

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

   5. Taylor expanded in c around 0 64.4%

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

      1. unpow2 64.4%

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

      2. *-lft-identity 64.4%

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

      3. times-frac 64.4%

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

      4. /-rgt-identity 64.4%

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

   7. Simplified 64.4%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{+43}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2.35 \cdot 10^{-60}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+85}:\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 12: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{+42}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{+86}:\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -7e+42)
   (/ b d)
   (if (<= d 5e-60)
     (+ (/ a c) (/ (/ b (/ c d)) c))
     (if (<= d 5.2e+86) (/ a (+ c (* d (/ d c)))) (/ b d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -7e+42) {
		tmp = b / d;
	} else if (d <= 5e-60) {
		tmp = (a / c) + ((b / (c / d)) / c);
	} else if (d <= 5.2e+86) {
		tmp = a / (c + (d * (d / 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 <= (-7d+42)) then
        tmp = b / d
    else if (d <= 5d-60) then
        tmp = (a / c) + ((b / (c / d)) / c)
    else if (d <= 5.2d+86) then
        tmp = a / (c + (d * (d / 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 <= -7e+42) {
		tmp = b / d;
	} else if (d <= 5e-60) {
		tmp = (a / c) + ((b / (c / d)) / c);
	} else if (d <= 5.2e+86) {
		tmp = a / (c + (d * (d / c)));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -7e+42:
		tmp = b / d
	elif d <= 5e-60:
		tmp = (a / c) + ((b / (c / d)) / c)
	elif d <= 5.2e+86:
		tmp = a / (c + (d * (d / c)))
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -7e+42)
		tmp = Float64(b / d);
	elseif (d <= 5e-60)
		tmp = Float64(Float64(a / c) + Float64(Float64(b / Float64(c / d)) / c));
	elseif (d <= 5.2e+86)
		tmp = Float64(a / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -7e+42)
		tmp = b / d;
	elseif (d <= 5e-60)
		tmp = (a / c) + ((b / (c / d)) / c);
	elseif (d <= 5.2e+86)
		tmp = a / (c + (d * (d / c)));
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -7e+42], N[(b / d), $MachinePrecision], If[LessEqual[d, 5e-60], N[(N[(a / c), $MachinePrecision] + N[(N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.2e+86], N[(a / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.00000000000000047e42 or 5.1999999999999995e86 < d

   1. Initial program 43.2%

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

   2. Taylor expanded in c around 0 72.9%

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

   if -7.00000000000000047e42 < d < 5.0000000000000001e-60

   1. Initial program 72.6%

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

   2. Taylor expanded in c around inf 73.1%

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

      1. unpow2 73.1%

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

   4. Simplified 73.1%

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

      1. clear-num 73.1%

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

      2. inv-pow 73.1%

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

      3. *-commutative 73.1%

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

      4. times-frac 78.2%

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

   6. Applied egg-rr 78.2%

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

      1. unpow-1 78.2%

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

      2. associate-*l/ 78.9%

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

   8. Simplified 78.9%

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

      1. clear-num 78.9%

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

      2. add-cube-cbrt 78.8%

         \[\leadsto \frac{a}{c} + \frac{\color{blue}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}}}{c \cdot \frac{c}{d}} \]

      3. times-frac 79.5%

         \[\leadsto \frac{a}{c} + \color{blue}{\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{c} \cdot \frac{\sqrt[3]{b}}{\frac{c}{d}}} \]

      4. pow2 79.5%

         \[\leadsto \frac{a}{c} + \frac{\color{blue}{{\left(\sqrt[3]{b}\right)}^{2}}}{c} \cdot \frac{\sqrt[3]{b}}{\frac{c}{d}} \]

   10. Applied egg-rr 79.5%

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

       1. associate-*l/ 79.5%

          \[\leadsto \frac{a}{c} + \color{blue}{\frac{{\left(\sqrt[3]{b}\right)}^{2} \cdot \frac{\sqrt[3]{b}}{\frac{c}{d}}}{c}} \]

       2. associate-*r/ 79.5%

          \[\leadsto \frac{a}{c} + \frac{\color{blue}{\frac{{\left(\sqrt[3]{b}\right)}^{2} \cdot \sqrt[3]{b}}{\frac{c}{d}}}}{c} \]

       3. unpow2 79.5%

          \[\leadsto \frac{a}{c} + \frac{\frac{\color{blue}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)} \cdot \sqrt[3]{b}}{\frac{c}{d}}}{c} \]

       4. rem-3cbrt-lft 79.6%

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

   12. Simplified 79.6%

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

   if 5.0000000000000001e-60 < d < 5.1999999999999995e86

   1. Initial program 74.5%

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

   2. Taylor expanded in a around inf 46.1%

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

      1. *-commutative 46.1%

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

      2. associate-/l* 46.7%

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

      3. unpow2 46.7%

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

      4. unpow2 46.7%

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

      5. +-commutative 46.7%

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

      6. fma-udef 46.8%

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

   4. Simplified 46.8%

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

   5. Taylor expanded in c around 0 64.4%

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

      1. unpow2 64.4%

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

      2. *-lft-identity 64.4%

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

      3. times-frac 64.4%

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

      4. /-rgt-identity 64.4%

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

   7. Simplified 64.4%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{+42}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{+86}:\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 13: 68.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -9.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 6.7 \cdot 10^{+84}:\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -9.6e+42)
   (/ b d)
   (if (<= d 6.7e+84) (/ a (+ c (* d (/ d c)))) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -9.6e+42) {
		tmp = b / d;
	} else if (d <= 6.7e+84) {
		tmp = a / (c + (d * (d / 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 <= (-9.6d+42)) then
        tmp = b / d
    else if (d <= 6.7d+84) then
        tmp = a / (c + (d * (d / 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 <= -9.6e+42) {
		tmp = b / d;
	} else if (d <= 6.7e+84) {
		tmp = a / (c + (d * (d / c)));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -9.6e+42:
		tmp = b / d
	elif d <= 6.7e+84:
		tmp = a / (c + (d * (d / c)))
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -9.6e+42)
		tmp = Float64(b / d);
	elseif (d <= 6.7e+84)
		tmp = Float64(a / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -9.6e+42)
		tmp = b / d;
	elseif (d <= 6.7e+84)
		tmp = a / (c + (d * (d / c)));
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -9.6e+42], N[(b / d), $MachinePrecision], If[LessEqual[d, 6.7e+84], N[(a / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -9.5999999999999994e42 or 6.70000000000000041e84 < d

   1. Initial program 43.2%

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

   2. Taylor expanded in c around 0 72.9%

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

   if -9.5999999999999994e42 < d < 6.70000000000000041e84

   1. Initial program 73.0%

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

   2. Taylor expanded in a around inf 44.2%

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

      1. *-commutative 44.2%

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

      2. associate-/l* 49.6%

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

      3. unpow2 49.6%

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

      4. unpow2 49.6%

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

      5. +-commutative 49.6%

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

      6. fma-udef 49.6%

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

   4. Simplified 49.6%

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

   5. Taylor expanded in c around 0 65.9%

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

      1. unpow2 65.9%

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

      2. *-lft-identity 65.9%

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

      3. times-frac 65.9%

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

      4. /-rgt-identity 65.9%

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

   7. Simplified 65.9%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 6.7 \cdot 10^{+84}:\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 14: 63.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.2 \cdot 10^{+104}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 8:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.2e+104) (/ a c) (if (<= c 8.0) (/ b d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.2e+104) {
		tmp = a / c;
	} else if (c <= 8.0) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-5.2d+104)) then
        tmp = a / c
    else if (c <= 8.0d0) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.2e+104) {
		tmp = a / c;
	} else if (c <= 8.0) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -5.2e+104:
		tmp = a / c
	elif c <= 8.0:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.2e+104)
		tmp = Float64(a / c);
	elseif (c <= 8.0)
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -5.2e+104)
		tmp = a / c;
	elseif (c <= 8.0)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.2e+104], N[(a / c), $MachinePrecision], If[LessEqual[c, 8.0], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -5.20000000000000001e104 or 8 < c

   1. Initial program 46.7%

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

   2. Taylor expanded in c around inf 66.7%

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

   if -5.20000000000000001e104 < c < 8

   1. Initial program 71.5%

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

   2. Taylor expanded in c around 0 61.6%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.2 \cdot 10^{+104}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 8:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 15: 41.8% 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 61.2%

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

2. Taylor expanded in c around inf 40.2%

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

3. Final simplification 40.2%

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

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 2023207 
(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))))