Complex division, imag part

Percentage Accurate: 60.7% → 97.3%

Time: 12.9s

Alternatives: 14

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{-a}{\mathsf{hypot}\left(c, d\right)}\right) \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (fma
  (/ c (hypot c d))
  (/ b (hypot c d))
  (* (/ d (hypot c d)) (/ (- a) (hypot c d)))))

C

double code(double a, double b, double c, double d) {
	return fma((c / hypot(c, d)), (b / hypot(c, d)), ((d / hypot(c, d)) * (-a / hypot(c, d))));
}

Julia

function code(a, b, c, d)
	return fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(Float64(d / hypot(c, d)) * Float64(Float64(-a) / hypot(c, d))))
end

Wolfram

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

Derivation

1. Initial program 62.2%

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

   1. div-sub 59.8%

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

   2. sub-neg 59.8%

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

   3. *-commutative 59.8%

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

   4. add-sqr-sqrt 59.8%

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

   5. times-frac 62.1%

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

   6. fma-def 62.1%

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

   7. hypot-def 62.1%

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

   8. hypot-def 74.9%

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

   9. associate-/l* 79.6%

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

   10. add-sqr-sqrt 79.6%

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

   11. pow2 79.6%

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

   12. hypot-def 79.6%

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

3. Applied egg-rr 79.6%

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

   1. div-inv 79.5%

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

   2. clear-num 79.6%

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

   3. add-sqr-sqrt 41.4%

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

   4. sqrt-unprod 54.1%

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

   5. sqr-neg 54.1%

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

   6. sqrt-unprod 24.0%

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

   7. add-sqr-sqrt 50.5%

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

   8. div-inv 50.5%

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

   9. associate-*l* 49.6%

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

   10. *-commutative 49.6%

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

   11. div-inv 49.6%

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

   12. unpow2 49.6%

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

   13. times-frac 52.1%

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

   14. add-sqr-sqrt 25.4%

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

   15. sqrt-unprod 61.5%

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

   16. sqr-neg 61.5%

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

   17. sqrt-unprod 51.4%

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

   18. add-sqr-sqrt 98.8%

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

5. Applied egg-rr 98.8%

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

6. Final simplification 98.8%

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

Alternative 2: 89.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;d \leq -3.7 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_1, \frac{-a}{d}\right)\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_1, \frac{-a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ c (hypot c d))) (t_1 (/ b (hypot c d))))
   (if (<= d -3.7e+161)
     (fma t_0 t_1 (/ (- a) d))
     (if (<= d 5e+140)
       (fma t_0 t_1 (/ (- a) (/ (pow (hypot c d) 2.0) d)))
       (/ (- (* b (/ c d)) a) (hypot c d))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = c / hypot(c, d);
	double t_1 = b / hypot(c, d);
	double tmp;
	if (d <= -3.7e+161) {
		tmp = fma(t_0, t_1, (-a / d));
	} else if (d <= 5e+140) {
		tmp = fma(t_0, t_1, (-a / (pow(hypot(c, d), 2.0) / d)));
	} else {
		tmp = ((b * (c / d)) - a) / hypot(c, d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(c / hypot(c, d))
	t_1 = Float64(b / hypot(c, d))
	tmp = 0.0
	if (d <= -3.7e+161)
		tmp = fma(t_0, t_1, Float64(Float64(-a) / d));
	elseif (d <= 5e+140)
		tmp = fma(t_0, t_1, Float64(Float64(-a) / Float64((hypot(c, d) ^ 2.0) / d)));
	else
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / hypot(c, d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.7e+161], N[(t$95$0 * t$95$1 + N[((-a) / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5e+140], N[(t$95$0 * t$95$1 + N[((-a) / N[(N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.69999999999999979e161

   1. Initial program 34.7%

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

      1. div-sub 34.7%

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

      2. sub-neg 34.7%

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

      3. *-commutative 34.7%

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

      4. add-sqr-sqrt 34.7%

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

      5. times-frac 34.8%

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

      6. fma-def 34.8%

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

      7. hypot-def 34.8%

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

      8. hypot-def 42.7%

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

      9. associate-/l* 47.3%

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

      10. add-sqr-sqrt 47.3%

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

      11. pow2 47.3%

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

      12. hypot-def 47.3%

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

   3. Applied egg-rr 47.3%

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

   4. Taylor expanded in c around 0 87.0%

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

   if -3.69999999999999979e161 < d < 5.00000000000000008e140

   1. Initial program 72.5%

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

      1. div-sub 69.3%

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

      2. sub-neg 69.3%

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

      3. *-commutative 69.3%

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

      4. add-sqr-sqrt 69.3%

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

      5. times-frac 72.3%

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

      6. fma-def 72.3%

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

      7. hypot-def 72.3%

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

      8. hypot-def 86.8%

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

      9. associate-/l* 92.1%

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

      10. add-sqr-sqrt 92.1%

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

      11. pow2 92.1%

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

      12. hypot-def 92.1%

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

   3. Applied egg-rr 92.1%

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

   if 5.00000000000000008e140 < d

   1. Initial program 33.6%

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

      1. *-un-lft-identity 33.6%

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

      2. add-sqr-sqrt 33.6%

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

      3. times-frac 33.6%

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

      4. hypot-def 33.6%

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

      5. hypot-def 55.5%

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

   3. Applied egg-rr 55.5%

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

   4. Taylor expanded in c around 0 84.8%

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

      1. neg-mul-1 84.8%

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

      2. +-commutative 84.8%

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

      3. unsub-neg 84.8%

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

      4. associate-/l* 96.8%

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

   6. Simplified 96.8%

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

      1. expm1-log1p-u 87.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\right)\right)} \]

      2. expm1-udef 38.4%

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

      3. associate-*l/ 38.4%

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

      4. *-un-lft-identity 38.4%

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

      5. div-inv 38.4%

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

      6. clear-num 38.4%

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

   8. Applied egg-rr 38.4%

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

      1. expm1-def 87.8%

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

      2. expm1-log1p 97.1%

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

   10. Simplified 97.1%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.7 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 3: 88.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := c \cdot b - d \cdot a\\ t_2 := \frac{t_1}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_0 \cdot \frac{c}{\frac{\mathsf{hypot}\left(c, d\right)}{b}} - d \cdot \frac{a}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+200}:\\ \;\;\;\;t_0 \cdot \frac{t_1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot c d)))
        (t_1 (- (* c b) (* d a)))
        (t_2 (/ t_1 (+ (* c c) (* d d)))))
   (if (<= t_2 (- INFINITY))
     (- (* t_0 (/ c (/ (hypot c d) b))) (* d (/ a (pow (hypot c d) 2.0))))
     (if (<= t_2 5e+200)
       (* t_0 (/ t_1 (hypot c d)))
       (fma (/ c (hypot c d)) (/ b (hypot c d)) (/ (- a) d))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = 1.0 / hypot(c, d);
	double t_1 = (c * b) - (d * a);
	double t_2 = t_1 / ((c * c) + (d * d));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (t_0 * (c / (hypot(c, d) / b))) - (d * (a / pow(hypot(c, d), 2.0)));
	} else if (t_2 <= 5e+200) {
		tmp = t_0 * (t_1 / hypot(c, d));
	} else {
		tmp = fma((c / hypot(c, d)), (b / hypot(c, d)), (-a / d));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(1.0 / hypot(c, d))
	t_1 = Float64(Float64(c * b) - Float64(d * a))
	t_2 = Float64(t_1 / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 * Float64(c / Float64(hypot(c, d) / b))) - Float64(d * Float64(a / (hypot(c, d) ^ 2.0))));
	elseif (t_2 <= 5e+200)
		tmp = Float64(t_0 * Float64(t_1 / hypot(c, d)));
	else
		tmp = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(Float64(-a) / d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(t$95$0 * N[(c / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(d * N[(a / N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+200], N[(t$95$0 * N[(t$95$1 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[((-a) / d), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 52.7%

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

      1. div-sub 33.6%

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

      2. sub-neg 33.6%

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

      3. *-un-lft-identity 33.6%

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

      4. add-sqr-sqrt 33.6%

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

      5. times-frac 33.6%

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

      6. fma-def 33.6%

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

      7. hypot-def 33.6%

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

      8. hypot-def 52.4%

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

      9. associate-/l* 69.8%

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

      10. add-sqr-sqrt 69.8%

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

      11. pow2 69.8%

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

      12. hypot-def 69.8%

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

   3. Applied egg-rr 69.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(c, d\right)}, \frac{b \cdot c}{\mathsf{hypot}\left(c, d\right)}, -\frac{a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)} \]
   4. Step-by-step derivation

      1. fma-neg 69.8%

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

      2. *-commutative 69.8%

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

      3. associate-/l* 93.5%

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

      4. associate-/r/ 93.5%

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

      5. *-commutative 93.5%

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

   5. Simplified 93.5%

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

   if -inf.0 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 5.00000000000000019e200

   1. Initial program 83.9%

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

      1. *-un-lft-identity 83.9%

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

      2. add-sqr-sqrt 83.9%

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

      3. times-frac 83.8%

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

      4. hypot-def 83.8%

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

      5. hypot-def 98.9%

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

   3. Applied egg-rr 98.9%

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

   if 5.00000000000000019e200 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 16.4%

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

      1. div-sub 12.3%

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

      2. sub-neg 12.3%

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

      3. *-commutative 12.3%

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

      4. add-sqr-sqrt 12.3%

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

      5. times-frac 14.8%

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

      6. fma-def 14.8%

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

      7. hypot-def 14.8%

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

      8. hypot-def 43.1%

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

      9. associate-/l* 55.1%

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

      10. add-sqr-sqrt 55.1%

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

      11. pow2 55.1%

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

      12. hypot-def 55.1%

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

   3. Applied egg-rr 55.1%

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

   4. Taylor expanded in c around 0 72.8%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq -\infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\frac{\mathsf{hypot}\left(c, d\right)}{b}} - d \cdot \frac{a}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\\ \mathbf{elif}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+200}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \end{array} \]

Alternative 4: 89.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot b - d \cdot a\\ t_1 := \frac{t_0}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{+200}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c b) (* d a))) (t_1 (/ t_0 (+ (* c c) (* d d)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e+200)))
     (fma (/ c (hypot c d)) (/ b (hypot c d)) (/ (- a) d))
     (* (/ 1.0 (hypot c d)) (/ t_0 (hypot c d))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * b) - (d * a);
	double t_1 = t_0 / ((c * c) + (d * d));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e+200)) {
		tmp = fma((c / hypot(c, d)), (b / hypot(c, d)), (-a / d));
	} else {
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * b) - Float64(d * a))
	t_1 = Float64(t_0 / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e+200))
		tmp = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(Float64(-a) / d));
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, d)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e+200]], $MachinePrecision]], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[((-a) / d), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < -inf.0 or 5.00000000000000019e200 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 22.8%

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

      1. div-sub 16.1%

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

      2. sub-neg 16.1%

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

      3. *-commutative 16.1%

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

      4. add-sqr-sqrt 16.1%

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

      5. times-frac 22.3%

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

      6. fma-def 22.3%

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

      7. hypot-def 22.3%

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

      8. hypot-def 48.9%

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

      9. associate-/l* 61.9%

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

      10. add-sqr-sqrt 61.9%

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

      11. pow2 61.9%

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

      12. hypot-def 61.9%

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

   3. Applied egg-rr 61.9%

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

   4. Taylor expanded in c around 0 75.5%

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

   if -inf.0 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 5.00000000000000019e200

   1. Initial program 83.9%

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

      1. *-un-lft-identity 83.9%

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

      2. add-sqr-sqrt 83.9%

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

      3. times-frac 83.8%

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

      4. hypot-def 83.8%

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

      5. hypot-def 98.9%

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

   3. Applied egg-rr 98.9%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq -\infty \lor \neg \left(\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+200}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 5: 85.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot b - d \cdot a\\ \mathbf{if}\;\frac{t_0}{c \cdot c + d \cdot d} \leq 10^{+230}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c b) (* d a))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 1e+230)
     (* (/ 1.0 (hypot c d)) (/ t_0 (hypot c d)))
     (* (/ 1.0 d) (- (/ b (/ d c)) a)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * b) - (d * a);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+230) {
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	} else {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (c * b) - (d * a);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+230) {
		tmp = (1.0 / Math.hypot(c, d)) * (t_0 / Math.hypot(c, d));
	} else {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (c * b) - (d * a)
	tmp = 0
	if (t_0 / ((c * c) + (d * d))) <= 1e+230:
		tmp = (1.0 / math.hypot(c, d)) * (t_0 / math.hypot(c, d))
	else:
		tmp = (1.0 / d) * ((b / (d / c)) - a)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * b) - Float64(d * a))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= 1e+230)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, d)));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(b / Float64(d / c)) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (c * b) - (d * a);
	tmp = 0.0;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+230)
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	else
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+230], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 1.0000000000000001e230

   1. Initial program 81.3%

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

      1. *-un-lft-identity 81.3%

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

      2. add-sqr-sqrt 81.3%

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

      3. times-frac 81.3%

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

      4. hypot-def 81.3%

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

      5. hypot-def 95.4%

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

   3. Applied egg-rr 95.4%

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

   if 1.0000000000000001e230 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 14.2%

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

      1. *-un-lft-identity 14.2%

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

      2. add-sqr-sqrt 14.2%

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

      3. times-frac 14.2%

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

      4. hypot-def 14.2%

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

      5. hypot-def 18.9%

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

   3. Applied egg-rr 18.9%

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

   4. Taylor expanded in c around 0 23.1%

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

      1. neg-mul-1 23.1%

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

      2. +-commutative 23.1%

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

      3. unsub-neg 23.1%

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

      4. associate-/l* 31.4%

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

   6. Simplified 31.4%

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

   7. Taylor expanded in c around 0 61.8%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 10^{+230}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \end{array} \]

Alternative 6: 79.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c \cdot \frac{c}{a}}\\ \mathbf{elif}\;c \leq 1.2 \cdot 10^{-133}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{+42}:\\ \;\;\;\;\left(c \cdot b - d \cdot a\right) \cdot {\left(\mathsf{hypot}\left(c, d\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.4e+29)
   (- (/ b c) (/ d (* c (/ c a))))
   (if (<= c 1.2e-133)
     (* (/ 1.0 d) (- (/ b (/ d c)) a))
     (if (<= c 1.7e+42)
       (* (- (* c b) (* d a)) (pow (hypot c d) -2.0))
       (- (/ b c) (* (/ d c) (/ a c)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.4e+29) {
		tmp = (b / c) - (d / (c * (c / a)));
	} else if (c <= 1.2e-133) {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	} else if (c <= 1.7e+42) {
		tmp = ((c * b) - (d * a)) * pow(hypot(c, d), -2.0);
	} else {
		tmp = (b / c) - ((d / c) * (a / c));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.4e+29) {
		tmp = (b / c) - (d / (c * (c / a)));
	} else if (c <= 1.2e-133) {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	} else if (c <= 1.7e+42) {
		tmp = ((c * b) - (d * a)) * Math.pow(Math.hypot(c, d), -2.0);
	} else {
		tmp = (b / c) - ((d / c) * (a / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -3.4e+29:
		tmp = (b / c) - (d / (c * (c / a)))
	elif c <= 1.2e-133:
		tmp = (1.0 / d) * ((b / (d / c)) - a)
	elif c <= 1.7e+42:
		tmp = ((c * b) - (d * a)) * math.pow(math.hypot(c, d), -2.0)
	else:
		tmp = (b / c) - ((d / c) * (a / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.4e+29)
		tmp = Float64(Float64(b / c) - Float64(d / Float64(c * Float64(c / a))));
	elseif (c <= 1.2e-133)
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(b / Float64(d / c)) - a));
	elseif (c <= 1.7e+42)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) * (hypot(c, d) ^ -2.0));
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(d / c) * Float64(a / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -3.4e+29)
		tmp = (b / c) - (d / (c * (c / a)));
	elseif (c <= 1.2e-133)
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	elseif (c <= 1.7e+42)
		tmp = ((c * b) - (d * a)) * (hypot(c, d) ^ -2.0);
	else
		tmp = (b / c) - ((d / c) * (a / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.4e+29], N[(N[(b / c), $MachinePrecision] - N[(d / N[(c * N[(c / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.2e-133], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.7e+42], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(N[(b / c), $MachinePrecision] - N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -3.39999999999999981e29

   1. Initial program 44.2%

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

   2. Taylor expanded in c around inf 70.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   3. Step-by-step derivation

      1. +-commutative 70.4%

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

      2. mul-1-neg 70.4%

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

      3. unsub-neg 70.4%

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

      4. associate-/l* 68.8%

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

      5. associate-/r/ 70.6%

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

   4. Simplified 70.6%

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

      1. *-un-lft-identity 70.6%

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

      2. unpow2 70.6%

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

      3. times-frac 77.7%

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

   6. Applied egg-rr 77.7%

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

      1. associate-*l/ 77.7%

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

      2. *-lft-identity 77.7%

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

   8. Simplified 77.7%

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

      1. *-commutative 77.7%

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

      2. clear-num 77.7%

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

      3. un-div-inv 77.7%

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

      4. div-inv 77.7%

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

      5. clear-num 77.8%

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

   10. Applied egg-rr 77.8%

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

   if -3.39999999999999981e29 < c < 1.2e-133

   1. Initial program 69.8%

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

      1. *-un-lft-identity 69.8%

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

      2. add-sqr-sqrt 69.8%

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

      3. times-frac 69.7%

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

      4. hypot-def 69.7%

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

      5. hypot-def 81.7%

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

   3. Applied egg-rr 81.7%

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

   4. Taylor expanded in c around 0 47.8%

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

      1. neg-mul-1 47.8%

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

      2. +-commutative 47.8%

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

      3. unsub-neg 47.8%

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

      4. associate-/l* 49.7%

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

   6. Simplified 49.7%

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

   7. Taylor expanded in c around 0 85.5%

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

   if 1.2e-133 < c < 1.69999999999999988e42

   1. Initial program 86.2%

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

      1. clear-num 86.2%

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

      2. associate-/r/ 86.0%

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

      3. add-sqr-sqrt 86.0%

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

      4. pow2 86.0%

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

      5. hypot-def 86.0%

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

   3. Applied egg-rr 86.0%

      \[\leadsto \color{blue}{\frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}} \cdot \left(b \cdot c - a \cdot d\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 84.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\right)} \cdot \left(b \cdot c - a \cdot d\right) \]

      2. expm1-udef 38.8%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)} - 1\right)} \cdot \left(b \cdot c - a \cdot d\right) \]

      3. pow-flip 38.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{\left(-2\right)}}\right)} - 1\right) \cdot \left(b \cdot c - a \cdot d\right) \]

      4. metadata-eval 38.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(c, d\right)\right)}^{\color{blue}{-2}}\right)} - 1\right) \cdot \left(b \cdot c - a \cdot d\right) \]

   5. Applied egg-rr 38.8%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(c, d\right)\right)}^{-2}\right)} - 1\right)} \cdot \left(b \cdot c - a \cdot d\right) \]
   6. Step-by-step derivation

      1. expm1-def 84.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(c, d\right)\right)}^{-2}\right)\right)} \cdot \left(b \cdot c - a \cdot d\right) \]

      2. expm1-log1p 86.4%

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

   7. Simplified 86.4%

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

   if 1.69999999999999988e42 < c

   1. Initial program 49.4%

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

   2. Taylor expanded in c around inf 73.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   3. Step-by-step derivation

      1. +-commutative 73.5%

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

      2. mul-1-neg 73.5%

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

      3. unsub-neg 73.5%

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

      4. associate-/l* 75.8%

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

      5. associate-/r/ 75.8%

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

   4. Simplified 75.8%

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

      1. *-un-lft-identity 75.8%

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

      2. unpow2 75.8%

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

      3. times-frac 82.5%

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

   6. Applied egg-rr 82.5%

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

      1. associate-*l/ 82.5%

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

      2. *-lft-identity 82.5%

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

   8. Simplified 82.5%

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

      1. associate-*l/ 85.6%

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

      2. clear-num 85.6%

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

   10. Applied egg-rr 85.6%

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

       1. associate-/r/ 85.6%

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

       2. *-commutative 85.6%

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

       3. associate-*r/ 85.6%

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

       4. *-rgt-identity 85.6%

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

       5. associate-*r/ 85.7%

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

   12. Simplified 85.7%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c \cdot \frac{c}{a}}\\ \mathbf{elif}\;c \leq 1.2 \cdot 10^{-133}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{+42}:\\ \;\;\;\;\left(c \cdot b - d \cdot a\right) \cdot {\left(\mathsf{hypot}\left(c, d\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \end{array} \]

Alternative 7: 79.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c \cdot \frac{c}{a}}\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{-131}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, c \cdot b\right)}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.4e+29)
   (- (/ b c) (/ d (* c (/ c a))))
   (if (<= c 1.15e-131)
     (* (/ 1.0 d) (- (/ b (/ d c)) a))
     (if (<= c 2.6e+44)
       (/ (fma (- d) a (* c b)) (+ (* c c) (* d d)))
       (- (/ b c) (* (/ d c) (/ a c)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.4e+29) {
		tmp = (b / c) - (d / (c * (c / a)));
	} else if (c <= 1.15e-131) {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	} else if (c <= 2.6e+44) {
		tmp = fma(-d, a, (c * b)) / ((c * c) + (d * d));
	} else {
		tmp = (b / c) - ((d / c) * (a / c));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.4e+29)
		tmp = Float64(Float64(b / c) - Float64(d / Float64(c * Float64(c / a))));
	elseif (c <= 1.15e-131)
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(b / Float64(d / c)) - a));
	elseif (c <= 2.6e+44)
		tmp = Float64(fma(Float64(-d), a, Float64(c * b)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(d / c) * Float64(a / c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.4e+29], N[(N[(b / c), $MachinePrecision] - N[(d / N[(c * N[(c / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.15e-131], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.6e+44], N[(N[((-d) * a + N[(c * b), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / c), $MachinePrecision] - N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -3.39999999999999981e29

   1. Initial program 44.2%

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

   2. Taylor expanded in c around inf 70.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   3. Step-by-step derivation

      1. +-commutative 70.4%

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

      2. mul-1-neg 70.4%

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

      3. unsub-neg 70.4%

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

      4. associate-/l* 68.8%

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

      5. associate-/r/ 70.6%

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

   4. Simplified 70.6%

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

      1. *-un-lft-identity 70.6%

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

      2. unpow2 70.6%

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

      3. times-frac 77.7%

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

   6. Applied egg-rr 77.7%

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

      1. associate-*l/ 77.7%

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

      2. *-lft-identity 77.7%

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

   8. Simplified 77.7%

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

      1. *-commutative 77.7%

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

      2. clear-num 77.7%

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

      3. un-div-inv 77.7%

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

      4. div-inv 77.7%

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

      5. clear-num 77.8%

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

   10. Applied egg-rr 77.8%

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

   if -3.39999999999999981e29 < c < 1.15000000000000011e-131

   1. Initial program 70.1%

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

      1. *-un-lft-identity 70.1%

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

      2. add-sqr-sqrt 70.1%

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

      3. times-frac 70.0%

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

      4. hypot-def 70.0%

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

      5. hypot-def 81.9%

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

   3. Applied egg-rr 81.9%

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

   4. Taylor expanded in c around 0 48.3%

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

      1. neg-mul-1 48.3%

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

      2. +-commutative 48.3%

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

      3. unsub-neg 48.3%

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

      4. associate-/l* 50.1%

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

   6. Simplified 50.1%

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

   7. Taylor expanded in c around 0 85.7%

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

   if 1.15000000000000011e-131 < c < 2.5999999999999999e44

   1. Initial program 85.9%

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

      1. sub-neg 85.9%

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

      2. +-commutative 85.9%

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

      3. *-commutative 85.9%

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

      4. distribute-lft-neg-in 85.9%

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

      5. fma-def 85.9%

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

   3. Applied egg-rr 85.9%

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

   if 2.5999999999999999e44 < c

   1. Initial program 49.4%

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

   2. Taylor expanded in c around inf 73.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   3. Step-by-step derivation

      1. +-commutative 73.5%

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

      2. mul-1-neg 73.5%

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

      3. unsub-neg 73.5%

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

      4. associate-/l* 75.8%

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

      5. associate-/r/ 75.8%

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

   4. Simplified 75.8%

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

      1. *-un-lft-identity 75.8%

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

      2. unpow2 75.8%

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

      3. times-frac 82.5%

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

   6. Applied egg-rr 82.5%

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

      1. associate-*l/ 82.5%

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

      2. *-lft-identity 82.5%

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

   8. Simplified 82.5%

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

      1. associate-*l/ 85.6%

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

      2. clear-num 85.6%

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

   10. Applied egg-rr 85.6%

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

       1. associate-/r/ 85.6%

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

       2. *-commutative 85.6%

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

       3. associate-*r/ 85.6%

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

       4. *-rgt-identity 85.6%

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

       5. associate-*r/ 85.7%

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

   12. Simplified 85.7%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c \cdot \frac{c}{a}}\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{-131}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, c \cdot b\right)}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \end{array} \]

Alternative 8: 79.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+29}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c \cdot \frac{c}{a}}\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-131}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \mathbf{elif}\;c \leq 1.06 \cdot 10^{+40}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.25e+29)
   (- (/ b c) (/ d (* c (/ c a))))
   (if (<= c 6.5e-131)
     (* (/ 1.0 d) (- (/ b (/ d c)) a))
     (if (<= c 1.06e+40)
       (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))
       (- (/ b c) (* (/ d c) (/ a c)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.25e+29) {
		tmp = (b / c) - (d / (c * (c / a)));
	} else if (c <= 6.5e-131) {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	} else if (c <= 1.06e+40) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else {
		tmp = (b / c) - ((d / c) * (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 <= (-1.25d+29)) then
        tmp = (b / c) - (d / (c * (c / a)))
    else if (c <= 6.5d-131) then
        tmp = (1.0d0 / d) * ((b / (d / c)) - a)
    else if (c <= 1.06d+40) then
        tmp = ((c * b) - (d * a)) / ((c * c) + (d * d))
    else
        tmp = (b / c) - ((d / c) * (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 <= -1.25e+29) {
		tmp = (b / c) - (d / (c * (c / a)));
	} else if (c <= 6.5e-131) {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	} else if (c <= 1.06e+40) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else {
		tmp = (b / c) - ((d / c) * (a / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.25e+29:
		tmp = (b / c) - (d / (c * (c / a)))
	elif c <= 6.5e-131:
		tmp = (1.0 / d) * ((b / (d / c)) - a)
	elif c <= 1.06e+40:
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d))
	else:
		tmp = (b / c) - ((d / c) * (a / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.25e+29)
		tmp = Float64(Float64(b / c) - Float64(d / Float64(c * Float64(c / a))));
	elseif (c <= 6.5e-131)
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(b / Float64(d / c)) - a));
	elseif (c <= 1.06e+40)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(d / c) * Float64(a / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.25e+29)
		tmp = (b / c) - (d / (c * (c / a)));
	elseif (c <= 6.5e-131)
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	elseif (c <= 1.06e+40)
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	else
		tmp = (b / c) - ((d / c) * (a / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.25e+29], N[(N[(b / c), $MachinePrecision] - N[(d / N[(c * N[(c / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6.5e-131], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.06e+40], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / c), $MachinePrecision] - N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.25e29

   1. Initial program 44.2%

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

   2. Taylor expanded in c around inf 70.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   3. Step-by-step derivation

      1. +-commutative 70.4%

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

      2. mul-1-neg 70.4%

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

      3. unsub-neg 70.4%

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

      4. associate-/l* 68.8%

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

      5. associate-/r/ 70.6%

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

   4. Simplified 70.6%

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

      1. *-un-lft-identity 70.6%

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

      2. unpow2 70.6%

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

      3. times-frac 77.7%

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

   6. Applied egg-rr 77.7%

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

      1. associate-*l/ 77.7%

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

      2. *-lft-identity 77.7%

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

   8. Simplified 77.7%

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

      1. *-commutative 77.7%

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

      2. clear-num 77.7%

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

      3. un-div-inv 77.7%

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

      4. div-inv 77.7%

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

      5. clear-num 77.8%

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

   10. Applied egg-rr 77.8%

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

   if -1.25e29 < c < 6.5000000000000002e-131

   1. Initial program 70.1%

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

      1. *-un-lft-identity 70.1%

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

      2. add-sqr-sqrt 70.1%

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

      3. times-frac 70.0%

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

      4. hypot-def 70.0%

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

      5. hypot-def 81.9%

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

   3. Applied egg-rr 81.9%

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

   4. Taylor expanded in c around 0 48.3%

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

      1. neg-mul-1 48.3%

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

      2. +-commutative 48.3%

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

      3. unsub-neg 48.3%

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

      4. associate-/l* 50.1%

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

   6. Simplified 50.1%

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

   7. Taylor expanded in c around 0 85.7%

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

   if 6.5000000000000002e-131 < c < 1.05999999999999996e40

   1. Initial program 85.9%

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

   if 1.05999999999999996e40 < c

   1. Initial program 49.4%

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

   2. Taylor expanded in c around inf 73.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   3. Step-by-step derivation

      1. +-commutative 73.5%

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

      2. mul-1-neg 73.5%

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

      3. unsub-neg 73.5%

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

      4. associate-/l* 75.8%

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

      5. associate-/r/ 75.8%

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

   4. Simplified 75.8%

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

      1. *-un-lft-identity 75.8%

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

      2. unpow2 75.8%

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

      3. times-frac 82.5%

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

   6. Applied egg-rr 82.5%

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

      1. associate-*l/ 82.5%

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

      2. *-lft-identity 82.5%

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

   8. Simplified 82.5%

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

      1. associate-*l/ 85.6%

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

      2. clear-num 85.6%

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

   10. Applied egg-rr 85.6%

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

       1. associate-/r/ 85.6%

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

       2. *-commutative 85.6%

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

       3. associate-*r/ 85.6%

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

       4. *-rgt-identity 85.6%

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

       5. associate-*r/ 85.7%

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

   12. Simplified 85.7%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+29}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c \cdot \frac{c}{a}}\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-131}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \mathbf{elif}\;c \leq 1.06 \cdot 10^{+40}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \end{array} \]

Alternative 9: 78.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.45 \cdot 10^{+29} \lor \neg \left(c \leq 2.4 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.45e+29) (not (<= c 2.4e+20)))
   (/ (- b (* d (/ a c))) c)
   (* (/ 1.0 d) (- (/ b (/ d c)) a))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.45e+29) || !(c <= 2.4e+20)) {
		tmp = (b - (d * (a / c))) / c;
	} else {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	}
	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 <= (-1.45d+29)) .or. (.not. (c <= 2.4d+20))) then
        tmp = (b - (d * (a / c))) / c
    else
        tmp = (1.0d0 / d) * ((b / (d / c)) - a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.45e+29) || !(c <= 2.4e+20)) {
		tmp = (b - (d * (a / c))) / c;
	} else {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.45e+29) or not (c <= 2.4e+20):
		tmp = (b - (d * (a / c))) / c
	else:
		tmp = (1.0 / d) * ((b / (d / c)) - a)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.45e+29) || !(c <= 2.4e+20))
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	else
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(b / Float64(d / c)) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.45e+29) || ~((c <= 2.4e+20)))
		tmp = (b - (d * (a / c))) / c;
	else
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.45e+29], N[Not[LessEqual[c, 2.4e+20]], $MachinePrecision]], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.45e29 or 2.4e20 < c

   1. Initial program 48.4%

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

   2. Taylor expanded in c around inf 71.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   3. Step-by-step derivation

      1. +-commutative 71.3%

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

      2. mul-1-neg 71.3%

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

      3. unsub-neg 71.3%

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

      4. associate-/l* 71.7%

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

      5. associate-/r/ 72.5%

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

   4. Simplified 72.5%

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

      1. *-un-lft-identity 72.5%

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

      2. unpow2 72.5%

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

      3. times-frac 79.0%

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

   6. Applied egg-rr 79.0%

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

      1. associate-*l/ 79.0%

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

      2. *-lft-identity 79.0%

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

   8. Simplified 79.0%

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

      1. associate-*l/ 80.5%

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

      2. sub-div 80.5%

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

   10. Applied egg-rr 80.5%

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

   if -1.45e29 < c < 2.4e20

   1. Initial program 74.2%

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

      1. *-un-lft-identity 74.2%

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

      2. add-sqr-sqrt 74.2%

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

      3. times-frac 74.1%

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

      4. hypot-def 74.1%

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

      5. hypot-def 84.2%

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

   3. Applied egg-rr 84.2%

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

   4. Taylor expanded in c around 0 49.3%

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

      1. neg-mul-1 49.3%

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

      2. +-commutative 49.3%

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

      3. unsub-neg 49.3%

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

      4. associate-/l* 50.7%

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

   6. Simplified 50.7%

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

   7. Taylor expanded in c around 0 82.4%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.45 \cdot 10^{+29} \lor \neg \left(c \leq 2.4 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \end{array} \]

Alternative 10: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.5 \cdot 10^{+29} \lor \neg \left(c \leq 9.5 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.5e+29) (not (<= c 9.5e+18)))
   (- (/ b c) (* (/ d c) (/ a c)))
   (* (/ 1.0 d) (- (/ b (/ d c)) a))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.5e+29) || !(c <= 9.5e+18)) {
		tmp = (b / c) - ((d / c) * (a / c));
	} else {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	}
	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 <= (-1.5d+29)) .or. (.not. (c <= 9.5d+18))) then
        tmp = (b / c) - ((d / c) * (a / c))
    else
        tmp = (1.0d0 / d) * ((b / (d / c)) - a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.5e+29) || !(c <= 9.5e+18)) {
		tmp = (b / c) - ((d / c) * (a / c));
	} else {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.5e+29) or not (c <= 9.5e+18):
		tmp = (b / c) - ((d / c) * (a / c))
	else:
		tmp = (1.0 / d) * ((b / (d / c)) - a)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.5e+29) || !(c <= 9.5e+18))
		tmp = Float64(Float64(b / c) - Float64(Float64(d / c) * Float64(a / c)));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(b / Float64(d / c)) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.5e+29) || ~((c <= 9.5e+18)))
		tmp = (b / c) - ((d / c) * (a / c));
	else
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.5e+29], N[Not[LessEqual[c, 9.5e+18]], $MachinePrecision]], N[(N[(b / c), $MachinePrecision] - N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.5e29 or 9.5e18 < c

   1. Initial program 48.4%

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

   2. Taylor expanded in c around inf 71.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   3. Step-by-step derivation

      1. +-commutative 71.3%

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

      2. mul-1-neg 71.3%

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

      3. unsub-neg 71.3%

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

      4. associate-/l* 71.7%

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

      5. associate-/r/ 72.5%

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

   4. Simplified 72.5%

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

      1. *-un-lft-identity 72.5%

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

      2. unpow2 72.5%

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

      3. times-frac 79.0%

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

   6. Applied egg-rr 79.0%

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

      1. associate-*l/ 79.0%

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

      2. *-lft-identity 79.0%

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

   8. Simplified 79.0%

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

      1. associate-*l/ 80.5%

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

      2. clear-num 80.5%

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

   10. Applied egg-rr 80.5%

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

       1. associate-/r/ 80.5%

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

       2. *-commutative 80.5%

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

       3. associate-*r/ 80.5%

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

       4. *-rgt-identity 80.5%

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

       5. associate-*r/ 80.6%

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

   12. Simplified 80.6%

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

   if -1.5e29 < c < 9.5e18

   1. Initial program 74.2%

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

      1. *-un-lft-identity 74.2%

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

      2. add-sqr-sqrt 74.2%

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

      3. times-frac 74.1%

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

      4. hypot-def 74.1%

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

      5. hypot-def 84.2%

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

   3. Applied egg-rr 84.2%

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

   4. Taylor expanded in c around 0 49.3%

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

      1. neg-mul-1 49.3%

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

      2. +-commutative 49.3%

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

      3. unsub-neg 49.3%

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

      4. associate-/l* 50.7%

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

   6. Simplified 50.7%

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

   7. Taylor expanded in c around 0 82.4%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.5 \cdot 10^{+29} \lor \neg \left(c \leq 9.5 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \end{array} \]

Alternative 11: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.35 \cdot 10^{+29}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c \cdot \frac{c}{a}}\\ \mathbf{elif}\;c \leq 2 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.35e+29)
   (- (/ b c) (/ d (* c (/ c a))))
   (if (<= c 2e+19)
     (* (/ 1.0 d) (- (/ b (/ d c)) a))
     (- (/ b c) (* (/ d c) (/ a c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.35e+29) {
		tmp = (b / c) - (d / (c * (c / a)));
	} else if (c <= 2e+19) {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	} else {
		tmp = (b / c) - ((d / c) * (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 <= (-1.35d+29)) then
        tmp = (b / c) - (d / (c * (c / a)))
    else if (c <= 2d+19) then
        tmp = (1.0d0 / d) * ((b / (d / c)) - a)
    else
        tmp = (b / c) - ((d / c) * (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 <= -1.35e+29) {
		tmp = (b / c) - (d / (c * (c / a)));
	} else if (c <= 2e+19) {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	} else {
		tmp = (b / c) - ((d / c) * (a / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.35e+29:
		tmp = (b / c) - (d / (c * (c / a)))
	elif c <= 2e+19:
		tmp = (1.0 / d) * ((b / (d / c)) - a)
	else:
		tmp = (b / c) - ((d / c) * (a / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.35e+29)
		tmp = Float64(Float64(b / c) - Float64(d / Float64(c * Float64(c / a))));
	elseif (c <= 2e+19)
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(b / Float64(d / c)) - a));
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(d / c) * Float64(a / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.35e+29)
		tmp = (b / c) - (d / (c * (c / a)));
	elseif (c <= 2e+19)
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	else
		tmp = (b / c) - ((d / c) * (a / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.35e+29], N[(N[(b / c), $MachinePrecision] - N[(d / N[(c * N[(c / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2e+19], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(N[(b / c), $MachinePrecision] - N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.35e29

   1. Initial program 44.2%

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

   2. Taylor expanded in c around inf 70.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   3. Step-by-step derivation

      1. +-commutative 70.4%

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

      2. mul-1-neg 70.4%

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

      3. unsub-neg 70.4%

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

      4. associate-/l* 68.8%

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

      5. associate-/r/ 70.6%

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

   4. Simplified 70.6%

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

      1. *-un-lft-identity 70.6%

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

      2. unpow2 70.6%

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

      3. times-frac 77.7%

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

   6. Applied egg-rr 77.7%

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

      1. associate-*l/ 77.7%

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

      2. *-lft-identity 77.7%

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

   8. Simplified 77.7%

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

      1. *-commutative 77.7%

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

      2. clear-num 77.7%

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

      3. un-div-inv 77.7%

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

      4. div-inv 77.7%

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

      5. clear-num 77.8%

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

   10. Applied egg-rr 77.8%

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

   if -1.35e29 < c < 2e19

   1. Initial program 74.2%

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

      1. *-un-lft-identity 74.2%

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

      2. add-sqr-sqrt 74.2%

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

      3. times-frac 74.1%

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

      4. hypot-def 74.1%

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

      5. hypot-def 84.2%

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

   3. Applied egg-rr 84.2%

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

   4. Taylor expanded in c around 0 49.3%

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

      1. neg-mul-1 49.3%

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

      2. +-commutative 49.3%

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

      3. unsub-neg 49.3%

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

      4. associate-/l* 50.7%

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

   6. Simplified 50.7%

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

   7. Taylor expanded in c around 0 82.4%

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

   if 2e19 < c

   1. Initial program 51.8%

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

   2. Taylor expanded in c around inf 72.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   3. Step-by-step derivation

      1. +-commutative 72.0%

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

      2. mul-1-neg 72.0%

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

      3. unsub-neg 72.0%

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

      4. associate-/l* 74.0%

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

      5. associate-/r/ 74.0%

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

   4. Simplified 74.0%

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

      1. *-un-lft-identity 74.0%

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

      2. unpow2 74.0%

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

      3. times-frac 80.0%

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

   6. Applied egg-rr 80.0%

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

      1. associate-*l/ 80.1%

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

      2. *-lft-identity 80.1%

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

   8. Simplified 80.1%

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

      1. associate-*l/ 82.8%

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

      2. clear-num 82.8%

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

   10. Applied egg-rr 82.8%

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

       1. associate-/r/ 82.8%

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

       2. *-commutative 82.8%

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

       3. associate-*r/ 82.8%

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

       4. *-rgt-identity 82.8%

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

       5. associate-*r/ 82.9%

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

   12. Simplified 82.9%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.35 \cdot 10^{+29}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c \cdot \frac{c}{a}}\\ \mathbf{elif}\;c \leq 2 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \end{array} \]

Alternative 12: 70.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -6.8 \cdot 10^{+108} \lor \neg \left(d \leq 55\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -6.8e+108) (not (<= d 55.0)))
   (/ (- a) d)
   (/ (- b (* d (/ a c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -6.8e+108) || !(d <= 55.0)) {
		tmp = -a / d;
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-6.8d+108)) .or. (.not. (d <= 55.0d0))) then
        tmp = -a / d
    else
        tmp = (b - (d * (a / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -6.8e+108) || !(d <= 55.0)) {
		tmp = -a / d;
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -6.8e+108) or not (d <= 55.0):
		tmp = -a / d
	else:
		tmp = (b - (d * (a / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -6.8e+108) || !(d <= 55.0))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -6.8e+108) || ~((d <= 55.0)))
		tmp = -a / d;
	else
		tmp = (b - (d * (a / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -6.8e+108], N[Not[LessEqual[d, 55.0]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -6.79999999999999992e108 or 55 < d

   1. Initial program 48.3%

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

   2. Taylor expanded in c around 0 73.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   3. Step-by-step derivation

      1. associate-*r/ 73.0%

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

      2. neg-mul-1 73.0%

         \[\leadsto \frac{\color{blue}{-a}}{d} \]

   4. Simplified 73.0%

      \[\leadsto \color{blue}{\frac{-a}{d}} \]

   if -6.79999999999999992e108 < d < 55

   1. Initial program 74.4%

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

   2. Taylor expanded in c around inf 73.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   3. Step-by-step derivation

      1. +-commutative 73.4%

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

      2. mul-1-neg 73.4%

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

      3. unsub-neg 73.4%

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

      4. associate-/l* 75.1%

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

      5. associate-/r/ 72.5%

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

   4. Simplified 72.5%

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

      1. *-un-lft-identity 72.5%

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

      2. unpow2 72.5%

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

      3. times-frac 75.3%

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

   6. Applied egg-rr 75.3%

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

      1. associate-*l/ 75.3%

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

      2. *-lft-identity 75.3%

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

   8. Simplified 75.3%

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

      1. associate-*l/ 78.9%

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

      2. sub-div 79.0%

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

   10. Applied egg-rr 79.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.8 \cdot 10^{+108} \lor \neg \left(d \leq 55\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]

Alternative 13: 63.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.6 \cdot 10^{+32} \lor \neg \left(c \leq 2.1 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.6e+32) (not (<= c 2.1e+18))) (/ b c) (/ (- a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.6e+32) || !(c <= 2.1e+18)) {
		tmp = b / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-2.6d+32)) .or. (.not. (c <= 2.1d+18))) then
        tmp = b / c
    else
        tmp = -a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.6e+32) || !(c <= 2.1e+18)) {
		tmp = b / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.6e+32) or not (c <= 2.1e+18):
		tmp = b / c
	else:
		tmp = -a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.6e+32) || !(c <= 2.1e+18))
		tmp = Float64(b / c);
	else
		tmp = Float64(Float64(-a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -2.6e+32) || ~((c <= 2.1e+18)))
		tmp = b / c;
	else
		tmp = -a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.6e+32], N[Not[LessEqual[c, 2.1e+18]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[((-a) / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.6000000000000002e32 or 2.1e18 < c

   1. Initial program 47.6%

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

   2. Taylor expanded in c around inf 69.5%

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

   if -2.6000000000000002e32 < c < 2.1e18

   1. Initial program 74.5%

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

   2. Taylor expanded in c around 0 69.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   3. Step-by-step derivation

      1. associate-*r/ 69.3%

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

      2. neg-mul-1 69.3%

         \[\leadsto \frac{\color{blue}{-a}}{d} \]

   4. Simplified 69.3%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.6 \cdot 10^{+32} \lor \neg \left(c \leq 2.1 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]

Alternative 14: 41.8% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.2%

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

2. Taylor expanded in c around inf 40.6%

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

3. Final simplification 40.6%

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

Developer target: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

Reproduce

herbie shell --seed 2023312 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

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

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