Complex division, imag part

Percentage Accurate: 62.0% → 96.3%

Time: 13.0s

Alternatives: 11

Speedup: 0.6×

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: 62.0% 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: 96.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.4 \cdot 10^{+235}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{c}}{\frac{c}{d}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{\mathsf{fma}\left(c, \frac{c}{d}, d\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.4e+235)
   (- (/ b c) (/ (/ a c) (/ c d)))
   (fma (/ c (hypot c d)) (/ b (hypot c d)) (/ (- a) (fma c (/ c d) d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.4e+235) {
		tmp = (b / c) - ((a / c) / (c / d));
	} else {
		tmp = fma((c / hypot(c, d)), (b / hypot(c, d)), (-a / fma(c, (c / d), d)));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.4e+235)
		tmp = Float64(Float64(b / c) - Float64(Float64(a / c) / Float64(c / d)));
	else
		tmp = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(Float64(-a) / fma(c, Float64(c / d), d)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.4e+235], N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] / N[(c / d), $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) / N[(c * N[(c / d), $MachinePrecision] + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.40000000000000013e235

   1. Initial program 32.7%

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

   3. Taylor expanded in c around inf 74.6%

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

      1. +-commutative 74.6%

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

      2. mul-1-neg 74.6%

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

      3. unsub-neg 74.6%

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

      4. associate-/l* 76.3%

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

      5. associate-/r/ 76.3%

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

   5. Simplified 76.3%

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

      1. *-un-lft-identity 76.3%

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

      2. pow2 76.3%

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

      3. times-frac 85.3%

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

   7. Applied egg-rr 85.3%

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

      1. associate-*l/ 85.4%

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

      2. *-lft-identity 85.4%

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

   9. Simplified 85.4%

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

       1. associate-*l/ 99.9%

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

   11. Applied egg-rr 99.9%

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

       1. associate-/l* 99.9%

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

   13. Simplified 99.9%

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

   if -1.40000000000000013e235 < c

   1. Initial program 57.7%

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

      1. div-sub 55.0%

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

      2. sub-neg 55.0%

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

      3. *-commutative 55.0%

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

      4. add-sqr-sqrt 55.0%

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

      5. times-frac 60.3%

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

      6. fma-def 60.3%

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

      7. hypot-def 60.3%

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

      8. hypot-def 75.4%

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

      9. associate-/l* 77.7%

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

      10. add-sqr-sqrt 77.7%

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

      11. pow2 77.7%

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

      12. hypot-def 77.7%

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

   4. Applied egg-rr 77.7%

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

   5. Taylor expanded in c around 0 93.2%

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

      1. +-commutative 93.2%

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

      2. unpow2 93.2%

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

      3. associate-*r/ 97.1%

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

      4. fma-def 97.1%

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

   7. Simplified 97.1%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.4 \cdot 10^{+235}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{c}}{\frac{c}{d}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{\mathsf{fma}\left(c, \frac{c}{d}, d\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot b - a \cdot d\\ t_1 := \frac{t_0}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{+237}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c b) (* a d))) (t_1 (/ t_0 (+ (* c c) (* d d)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e+237)))
     (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) - (a * d);
	double t_1 = t_0 / ((c * c) + (d * d));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e+237)) {
		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(a * d))
	t_1 = Float64(t_0 / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e+237))
		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[(a * d), $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+237]], $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 5.0000000000000002e237 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 19.1%

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

      1. div-sub 12.1%

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

      2. sub-neg 12.1%

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

      3. *-commutative 12.1%

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

      4. add-sqr-sqrt 12.1%

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

      5. times-frac 27.1%

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

      6. fma-def 27.1%

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

      7. hypot-def 27.1%

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

      8. hypot-def 54.4%

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

      9. associate-/l* 61.4%

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

      10. add-sqr-sqrt 61.4%

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

      11. pow2 61.4%

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

      12. hypot-def 61.4%

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

   4. Applied egg-rr 61.4%

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

   5. Taylor expanded in c around 0 70.9%

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

   if -inf.0 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 5.0000000000000002e237

   1. Initial program 75.1%

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

      1. *-un-lft-identity 75.1%

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

      2. add-sqr-sqrt 75.1%

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

      3. times-frac 75.2%

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

      4. hypot-def 75.2%

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

      5. hypot-def 99.1%

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

   4. Applied egg-rr 99.1%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d} \leq -\infty \lor \neg \left(\frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+237}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot b - a \cdot d\\ \mathbf{if}\;\frac{t_0}{c \cdot c + d \cdot d} \leq 4 \cdot 10^{+302}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{a}{c} \cdot \frac{d}{c}, \frac{b}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c b) (* a d))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 4e+302)
     (* (/ 1.0 (hypot c d)) (/ t_0 (hypot c d)))
     (fma -1.0 (* (/ a c) (/ d c)) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * b) - (a * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 4e+302) {
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	} else {
		tmp = fma(-1.0, ((a / c) * (d / c)), (b / c));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * b) - Float64(a * d))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= 4e+302)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, d)));
	else
		tmp = fma(-1.0, Float64(Float64(a / c) * Float64(d / c)), Float64(b / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+302], 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[(-1.0 * N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision] + N[(b / c), $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))) < 4.0000000000000003e302

   1. Initial program 71.5%

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

      1. *-un-lft-identity 71.5%

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

      2. add-sqr-sqrt 71.5%

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

      3. times-frac 71.5%

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

      4. hypot-def 71.5%

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

      5. hypot-def 94.6%

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

   4. Applied egg-rr 94.6%

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

   if 4.0000000000000003e302 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 11.0%

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

   3. Taylor expanded in c around inf 41.6%

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

      1. fma-def 41.6%

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

      2. associate-/l* 44.3%

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

   5. Simplified 44.3%

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

      1. associate-/r/ 44.3%

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

      2. associate-*l/ 41.6%

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

      3. *-commutative 41.6%

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

      4. pow2 41.6%

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

      5. times-frac 54.1%

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

   7. Applied egg-rr 54.1%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d} \leq 4 \cdot 10^{+302}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{a}{c} \cdot \frac{d}{c}, \frac{b}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -8 \cdot 10^{+81}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{c}}{\frac{c}{d}}\\ \mathbf{elif}\;c \leq -4.8 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{-218}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 0.28:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* a d)) (+ (* c c) (* d d)))))
   (if (<= c -8e+81)
     (- (/ b c) (/ (/ a c) (/ c d)))
     (if (<= c -4.8e-156)
       t_0
       (if (<= c 2.2e-218)
         (/ (- a) d)
         (if (<= c 0.28) t_0 (* (/ c (hypot c d)) (/ b (hypot c d)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -8e+81) {
		tmp = (b / c) - ((a / c) / (c / d));
	} else if (c <= -4.8e-156) {
		tmp = t_0;
	} else if (c <= 2.2e-218) {
		tmp = -a / d;
	} else if (c <= 0.28) {
		tmp = t_0;
	} else {
		tmp = (c / hypot(c, d)) * (b / hypot(c, d));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -8e+81) {
		tmp = (b / c) - ((a / c) / (c / d));
	} else if (c <= -4.8e-156) {
		tmp = t_0;
	} else if (c <= 2.2e-218) {
		tmp = -a / d;
	} else if (c <= 0.28) {
		tmp = t_0;
	} else {
		tmp = (c / Math.hypot(c, d)) * (b / Math.hypot(c, d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (a * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -8e+81:
		tmp = (b / c) - ((a / c) / (c / d))
	elif c <= -4.8e-156:
		tmp = t_0
	elif c <= 2.2e-218:
		tmp = -a / d
	elif c <= 0.28:
		tmp = t_0
	else:
		tmp = (c / math.hypot(c, d)) * (b / math.hypot(c, d))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -8e+81)
		tmp = Float64(Float64(b / c) - Float64(Float64(a / c) / Float64(c / d)));
	elseif (c <= -4.8e-156)
		tmp = t_0;
	elseif (c <= 2.2e-218)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 0.28)
		tmp = t_0;
	else
		tmp = Float64(Float64(c / hypot(c, d)) * Float64(b / hypot(c, d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (a * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -8e+81)
		tmp = (b / c) - ((a / c) / (c / d));
	elseif (c <= -4.8e-156)
		tmp = t_0;
	elseif (c <= 2.2e-218)
		tmp = -a / d;
	elseif (c <= 0.28)
		tmp = t_0;
	else
		tmp = (c / hypot(c, d)) * (b / hypot(c, d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -8e+81], N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -4.8e-156], t$95$0, If[LessEqual[c, 2.2e-218], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 0.28], t$95$0, N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -7.99999999999999937e81

   1. Initial program 32.6%

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

   3. Taylor expanded in c around inf 70.3%

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

      1. +-commutative 70.3%

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

      2. mul-1-neg 70.3%

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

      3. unsub-neg 70.3%

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

      4. associate-/l* 69.7%

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

      5. associate-/r/ 71.2%

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

   5. Simplified 71.2%

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

      1. *-un-lft-identity 71.2%

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

      2. pow2 71.2%

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

      3. times-frac 79.6%

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

   7. Applied egg-rr 79.6%

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

      1. associate-*l/ 79.6%

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

      2. *-lft-identity 79.6%

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

   9. Simplified 79.6%

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

       1. associate-*l/ 84.9%

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

   11. Applied egg-rr 84.9%

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

       1. associate-/l* 85.0%

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

   13. Simplified 85.0%

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

   if -7.99999999999999937e81 < c < -4.8e-156 or 2.20000000000000007e-218 < c < 0.28000000000000003

   1. Initial program 81.2%

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

   if -4.8e-156 < c < 2.20000000000000007e-218

   1. Initial program 64.2%

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

   3. Taylor expanded in c around 0 88.1%

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

      1. associate-*r/ 88.1%

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

      2. neg-mul-1 88.1%

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

   5. Simplified 88.1%

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

   if 0.28000000000000003 < c

   1. Initial program 29.4%

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

      1. flip-- 18.4%

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

      2. clear-num 18.4%

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

      3. fma-def 18.4%

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

      4. pow2 18.4%

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

      5. pow2 18.4%

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

   4. Applied egg-rr 18.4%

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

   5. Taylor expanded in b around inf 30.0%

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

      1. associate-/r* 30.0%

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

   7. Simplified 30.0%

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

      1. associate-/r/ 30.0%

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

      2. inv-pow 30.0%

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

      3. pow-flip 30.0%

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

      4. metadata-eval 30.0%

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

      5. pow1 30.0%

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

      6. add-sqr-sqrt 30.0%

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

      7. hypot-udef 30.0%

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

      8. hypot-udef 30.0%

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

      9. times-frac 77.6%

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

   9. Applied egg-rr 77.6%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8 \cdot 10^{+81}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{c}}{\frac{c}{d}}\\ \mathbf{elif}\;c \leq -4.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{-218}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 0.28:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.9 \cdot 10^{+78}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{c}}{\frac{c}{d}}\\ \mathbf{elif}\;c \leq -8 \cdot 10^{-158}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 3.9 \cdot 10^{-218}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 4.3 \cdot 10^{+21}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{a}{c} \cdot \frac{d}{c}, \frac{b}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* a d)) (+ (* c c) (* d d)))))
   (if (<= c -1.9e+78)
     (- (/ b c) (/ (/ a c) (/ c d)))
     (if (<= c -8e-158)
       t_0
       (if (<= c 3.9e-218)
         (/ (- a) d)
         (if (<= c 4.3e+21) t_0 (fma -1.0 (* (/ a c) (/ d c)) (/ b c))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1.9e+78) {
		tmp = (b / c) - ((a / c) / (c / d));
	} else if (c <= -8e-158) {
		tmp = t_0;
	} else if (c <= 3.9e-218) {
		tmp = -a / d;
	} else if (c <= 4.3e+21) {
		tmp = t_0;
	} else {
		tmp = fma(-1.0, ((a / c) * (d / c)), (b / c));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -1.9e+78)
		tmp = Float64(Float64(b / c) - Float64(Float64(a / c) / Float64(c / d)));
	elseif (c <= -8e-158)
		tmp = t_0;
	elseif (c <= 3.9e-218)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 4.3e+21)
		tmp = t_0;
	else
		tmp = fma(-1.0, Float64(Float64(a / c) * Float64(d / c)), Float64(b / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.9e+78], N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -8e-158], t$95$0, If[LessEqual[c, 3.9e-218], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 4.3e+21], t$95$0, N[(-1.0 * N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.9e78

   1. Initial program 32.6%

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

   3. Taylor expanded in c around inf 70.3%

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

      1. +-commutative 70.3%

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

      2. mul-1-neg 70.3%

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

      3. unsub-neg 70.3%

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

      4. associate-/l* 69.7%

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

      5. associate-/r/ 71.2%

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

   5. Simplified 71.2%

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

      1. *-un-lft-identity 71.2%

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

      2. pow2 71.2%

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

      3. times-frac 79.6%

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

   7. Applied egg-rr 79.6%

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

      1. associate-*l/ 79.6%

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

      2. *-lft-identity 79.6%

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

   9. Simplified 79.6%

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

       1. associate-*l/ 84.9%

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

   11. Applied egg-rr 84.9%

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

       1. associate-/l* 85.0%

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

   13. Simplified 85.0%

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

   if -1.9e78 < c < -8.00000000000000052e-158 or 3.9e-218 < c < 4.3e21

   1. Initial program 80.6%

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

   if -8.00000000000000052e-158 < c < 3.9e-218

   1. Initial program 64.2%

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

   3. Taylor expanded in c around 0 88.1%

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

      1. associate-*r/ 88.1%

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

      2. neg-mul-1 88.1%

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

   5. Simplified 88.1%

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

   if 4.3e21 < c

   1. Initial program 28.6%

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

   3. Taylor expanded in c around inf 62.1%

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

      1. fma-def 62.1%

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

      2. associate-/l* 62.5%

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

   5. Simplified 62.5%

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

      1. associate-/r/ 64.3%

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

      2. associate-*l/ 62.1%

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

      3. *-commutative 62.1%

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

      4. pow2 62.1%

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

      5. times-frac 73.1%

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

   7. Applied egg-rr 73.1%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.9 \cdot 10^{+78}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{c}}{\frac{c}{d}}\\ \mathbf{elif}\;c \leq -8 \cdot 10^{-158}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 3.9 \cdot 10^{-218}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 4.3 \cdot 10^{+21}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{a}{c} \cdot \frac{d}{c}, \frac{b}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{c} - \frac{\frac{a}{c}}{\frac{c}{d}}\\ \mathbf{if}\;c \leq -6.8 \cdot 10^{+78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -4.3 \cdot 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-218}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 4.3 \cdot 10^{+21}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* a d)) (+ (* c c) (* d d))))
        (t_1 (- (/ b c) (/ (/ a c) (/ c d)))))
   (if (<= c -6.8e+78)
     t_1
     (if (<= c -4.3e-160)
       t_0
       (if (<= c 6.5e-218) (/ (- a) d) (if (<= c 4.3e+21) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (a * d)) / ((c * c) + (d * d));
	double t_1 = (b / c) - ((a / c) / (c / d));
	double tmp;
	if (c <= -6.8e+78) {
		tmp = t_1;
	} else if (c <= -4.3e-160) {
		tmp = t_0;
	} else if (c <= 6.5e-218) {
		tmp = -a / d;
	} else if (c <= 4.3e+21) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((c * b) - (a * d)) / ((c * c) + (d * d))
    t_1 = (b / c) - ((a / c) / (c / d))
    if (c <= (-6.8d+78)) then
        tmp = t_1
    else if (c <= (-4.3d-160)) then
        tmp = t_0
    else if (c <= 6.5d-218) then
        tmp = -a / d
    else if (c <= 4.3d+21) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (a * d)) / ((c * c) + (d * d));
	double t_1 = (b / c) - ((a / c) / (c / d));
	double tmp;
	if (c <= -6.8e+78) {
		tmp = t_1;
	} else if (c <= -4.3e-160) {
		tmp = t_0;
	} else if (c <= 6.5e-218) {
		tmp = -a / d;
	} else if (c <= 4.3e+21) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (a * d)) / ((c * c) + (d * d))
	t_1 = (b / c) - ((a / c) / (c / d))
	tmp = 0
	if c <= -6.8e+78:
		tmp = t_1
	elif c <= -4.3e-160:
		tmp = t_0
	elif c <= 6.5e-218:
		tmp = -a / d
	elif c <= 4.3e+21:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b / c) - Float64(Float64(a / c) / Float64(c / d)))
	tmp = 0.0
	if (c <= -6.8e+78)
		tmp = t_1;
	elseif (c <= -4.3e-160)
		tmp = t_0;
	elseif (c <= 6.5e-218)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 4.3e+21)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (a * d)) / ((c * c) + (d * d));
	t_1 = (b / c) - ((a / c) / (c / d));
	tmp = 0.0;
	if (c <= -6.8e+78)
		tmp = t_1;
	elseif (c <= -4.3e-160)
		tmp = t_0;
	elseif (c <= 6.5e-218)
		tmp = -a / d;
	elseif (c <= 4.3e+21)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -6.8e+78], t$95$1, If[LessEqual[c, -4.3e-160], t$95$0, If[LessEqual[c, 6.5e-218], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 4.3e+21], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -6.80000000000000014e78 or 4.3e21 < c

   1. Initial program 30.7%

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

   3. Taylor expanded in c around inf 66.3%

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

      1. +-commutative 66.3%

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

      2. mul-1-neg 66.3%

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

      3. unsub-neg 66.3%

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

      4. associate-/l* 66.2%

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

      5. associate-/r/ 67.9%

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

   5. Simplified 67.9%

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

      1. *-un-lft-identity 67.9%

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

      2. pow2 67.9%

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

      3. times-frac 75.6%

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

   7. Applied egg-rr 75.6%

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

      1. associate-*l/ 75.6%

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

      2. *-lft-identity 75.6%

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

   9. Simplified 75.6%

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

       1. associate-*l/ 79.2%

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

   11. Applied egg-rr 79.2%

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

       1. associate-/l* 79.3%

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

   13. Simplified 79.3%

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

   if -6.80000000000000014e78 < c < -4.30000000000000014e-160 or 6.49999999999999983e-218 < c < 4.3e21

   1. Initial program 80.6%

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

   if -4.30000000000000014e-160 < c < 6.49999999999999983e-218

   1. Initial program 64.2%

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

   3. Taylor expanded in c around 0 88.1%

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

      1. associate-*r/ 88.1%

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

      2. neg-mul-1 88.1%

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

   5. Simplified 88.1%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{c}}{\frac{c}{d}}\\ \mathbf{elif}\;c \leq -4.3 \cdot 10^{-160}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-218}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 4.3 \cdot 10^{+21}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{c}}{\frac{c}{d}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.8 \cdot 10^{-33} \lor \neg \left(c \leq 2.3 \cdot 10^{-92}\right) \land \left(c \leq 1.55 \cdot 10^{-44} \lor \neg \left(c \leq 4.4 \cdot 10^{+21}\right)\right):\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.8e-33)
         (and (not (<= c 2.3e-92)) (or (<= c 1.55e-44) (not (<= c 4.4e+21)))))
   (- (/ b c) (* d (/ (/ a c) c)))
   (/ (- a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.8e-33) || (!(c <= 2.3e-92) && ((c <= 1.55e-44) || !(c <= 4.4e+21)))) {
		tmp = (b / c) - (d * ((a / c) / 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 <= (-1.8d-33)) .or. (.not. (c <= 2.3d-92)) .and. (c <= 1.55d-44) .or. (.not. (c <= 4.4d+21))) then
        tmp = (b / c) - (d * ((a / c) / 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 <= -1.8e-33) || (!(c <= 2.3e-92) && ((c <= 1.55e-44) || !(c <= 4.4e+21)))) {
		tmp = (b / c) - (d * ((a / c) / c));
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.8e-33) or (not (c <= 2.3e-92) and ((c <= 1.55e-44) or not (c <= 4.4e+21))):
		tmp = (b / c) - (d * ((a / c) / c))
	else:
		tmp = -a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.8e-33) || (!(c <= 2.3e-92) && ((c <= 1.55e-44) || !(c <= 4.4e+21))))
		tmp = Float64(Float64(b / c) - Float64(d * Float64(Float64(a / c) / 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 <= -1.8e-33) || (~((c <= 2.3e-92)) && ((c <= 1.55e-44) || ~((c <= 4.4e+21)))))
		tmp = (b / c) - (d * ((a / c) / c));
	else
		tmp = -a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.8e-33], And[N[Not[LessEqual[c, 2.3e-92]], $MachinePrecision], Or[LessEqual[c, 1.55e-44], N[Not[LessEqual[c, 4.4e+21]], $MachinePrecision]]]], N[(N[(b / c), $MachinePrecision] - N[(d * N[(N[(a / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-a) / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.80000000000000017e-33 or 2.30000000000000016e-92 < c < 1.54999999999999992e-44 or 4.4e21 < c

   1. Initial program 44.5%

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

   3. Taylor expanded in c around inf 66.9%

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

      1. +-commutative 66.9%

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

      2. mul-1-neg 66.9%

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

      3. unsub-neg 66.9%

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

      4. associate-/l* 66.9%

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

      5. associate-/r/ 68.1%

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

   5. Simplified 68.1%

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

      1. *-un-lft-identity 68.1%

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

      2. pow2 68.1%

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

      3. times-frac 73.9%

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

   7. Applied egg-rr 73.9%

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

      1. associate-*l/ 73.9%

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

      2. *-lft-identity 73.9%

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

   9. Simplified 73.9%

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

   if -1.80000000000000017e-33 < c < 2.30000000000000016e-92 or 1.54999999999999992e-44 < c < 4.4e21

   1. Initial program 70.9%

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

   3. Taylor expanded in c around 0 74.5%

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

      1. associate-*r/ 74.5%

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

      2. neg-mul-1 74.5%

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

   5. Simplified 74.5%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.8 \cdot 10^{-33} \lor \neg \left(c \leq 2.3 \cdot 10^{-92}\right) \land \left(c \leq 1.55 \cdot 10^{-44} \lor \neg \left(c \leq 4.4 \cdot 10^{+21}\right)\right):\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.85 \cdot 10^{-38} \lor \neg \left(c \leq 2.3 \cdot 10^{-92} \lor \neg \left(c \leq 2.15 \cdot 10^{-42}\right) \land c \leq 1.15 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{c}}{\frac{c}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.85e-38)
         (not (or (<= c 2.3e-92) (and (not (<= c 2.15e-42)) (<= c 1.15e+22)))))
   (- (/ b c) (/ (/ a c) (/ c d)))
   (/ (- a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.85e-38) || !((c <= 2.3e-92) || (!(c <= 2.15e-42) && (c <= 1.15e+22)))) {
		tmp = (b / c) - ((a / c) / (c / d));
	} 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 <= (-1.85d-38)) .or. (.not. (c <= 2.3d-92) .or. (.not. (c <= 2.15d-42)) .and. (c <= 1.15d+22))) then
        tmp = (b / c) - ((a / c) / (c / d))
    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 <= -1.85e-38) || !((c <= 2.3e-92) || (!(c <= 2.15e-42) && (c <= 1.15e+22)))) {
		tmp = (b / c) - ((a / c) / (c / d));
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.85e-38) or not ((c <= 2.3e-92) or (not (c <= 2.15e-42) and (c <= 1.15e+22))):
		tmp = (b / c) - ((a / c) / (c / d))
	else:
		tmp = -a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.85e-38) || !((c <= 2.3e-92) || (!(c <= 2.15e-42) && (c <= 1.15e+22))))
		tmp = Float64(Float64(b / c) - Float64(Float64(a / c) / Float64(c / d)));
	else
		tmp = Float64(Float64(-a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.85e-38) || ~(((c <= 2.3e-92) || (~((c <= 2.15e-42)) && (c <= 1.15e+22)))))
		tmp = (b / c) - ((a / c) / (c / d));
	else
		tmp = -a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.85e-38], N[Not[Or[LessEqual[c, 2.3e-92], And[N[Not[LessEqual[c, 2.15e-42]], $MachinePrecision], LessEqual[c, 1.15e+22]]]], $MachinePrecision]], N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-a) / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.85e-38 or 2.30000000000000016e-92 < c < 2.1500000000000001e-42 or 1.1500000000000001e22 < c

   1. Initial program 44.5%

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

   3. Taylor expanded in c around inf 66.9%

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

      1. +-commutative 66.9%

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

      2. mul-1-neg 66.9%

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

      3. unsub-neg 66.9%

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

      4. associate-/l* 66.9%

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

      5. associate-/r/ 68.1%

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

   5. Simplified 68.1%

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

      1. *-un-lft-identity 68.1%

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

      2. pow2 68.1%

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

      3. times-frac 73.9%

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

   7. Applied egg-rr 73.9%

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

      1. associate-*l/ 73.9%

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

      2. *-lft-identity 73.9%

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

   9. Simplified 73.9%

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

       1. associate-*l/ 76.5%

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

   11. Applied egg-rr 76.5%

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

       1. associate-/l* 76.6%

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

   13. Simplified 76.6%

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

   if -1.85e-38 < c < 2.30000000000000016e-92 or 2.1500000000000001e-42 < c < 1.1500000000000001e22

   1. Initial program 70.9%

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

   3. Taylor expanded in c around 0 74.5%

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

      1. associate-*r/ 74.5%

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

      2. neg-mul-1 74.5%

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

   5. Simplified 74.5%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.85 \cdot 10^{-38} \lor \neg \left(c \leq 2.3 \cdot 10^{-92} \lor \neg \left(c \leq 2.15 \cdot 10^{-42}\right) \land c \leq 1.15 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{c}}{\frac{c}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ t_1 := \frac{b}{c} - \frac{\frac{a}{c}}{\frac{c}{d}}\\ \mathbf{if}\;c \leq -1.85 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 5.7 \cdot 10^{-95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 10^{-44}:\\ \;\;\;\;\frac{b}{c} + d \cdot \left(\frac{a}{c} \cdot \frac{-1}{c}\right)\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{+21}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)) (t_1 (- (/ b c) (/ (/ a c) (/ c d)))))
   (if (<= c -1.85e-34)
     t_1
     (if (<= c 5.7e-95)
       t_0
       (if (<= c 1e-44)
         (+ (/ b c) (* d (* (/ a c) (/ -1.0 c))))
         (if (<= c 4.4e+21) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double t_1 = (b / c) - ((a / c) / (c / d));
	double tmp;
	if (c <= -1.85e-34) {
		tmp = t_1;
	} else if (c <= 5.7e-95) {
		tmp = t_0;
	} else if (c <= 1e-44) {
		tmp = (b / c) + (d * ((a / c) * (-1.0 / c)));
	} else if (c <= 4.4e+21) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -a / d
    t_1 = (b / c) - ((a / c) / (c / d))
    if (c <= (-1.85d-34)) then
        tmp = t_1
    else if (c <= 5.7d-95) then
        tmp = t_0
    else if (c <= 1d-44) then
        tmp = (b / c) + (d * ((a / c) * ((-1.0d0) / c)))
    else if (c <= 4.4d+21) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double t_1 = (b / c) - ((a / c) / (c / d));
	double tmp;
	if (c <= -1.85e-34) {
		tmp = t_1;
	} else if (c <= 5.7e-95) {
		tmp = t_0;
	} else if (c <= 1e-44) {
		tmp = (b / c) + (d * ((a / c) * (-1.0 / c)));
	} else if (c <= 4.4e+21) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	t_1 = (b / c) - ((a / c) / (c / d))
	tmp = 0
	if c <= -1.85e-34:
		tmp = t_1
	elif c <= 5.7e-95:
		tmp = t_0
	elif c <= 1e-44:
		tmp = (b / c) + (d * ((a / c) * (-1.0 / c)))
	elif c <= 4.4e+21:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	t_1 = Float64(Float64(b / c) - Float64(Float64(a / c) / Float64(c / d)))
	tmp = 0.0
	if (c <= -1.85e-34)
		tmp = t_1;
	elseif (c <= 5.7e-95)
		tmp = t_0;
	elseif (c <= 1e-44)
		tmp = Float64(Float64(b / c) + Float64(d * Float64(Float64(a / c) * Float64(-1.0 / c))));
	elseif (c <= 4.4e+21)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -a / d;
	t_1 = (b / c) - ((a / c) / (c / d));
	tmp = 0.0;
	if (c <= -1.85e-34)
		tmp = t_1;
	elseif (c <= 5.7e-95)
		tmp = t_0;
	elseif (c <= 1e-44)
		tmp = (b / c) + (d * ((a / c) * (-1.0 / c)));
	elseif (c <= 4.4e+21)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.85e-34], t$95$1, If[LessEqual[c, 5.7e-95], t$95$0, If[LessEqual[c, 1e-44], N[(N[(b / c), $MachinePrecision] + N[(d * N[(N[(a / c), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4.4e+21], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.84999999999999994e-34 or 4.4e21 < c

   1. Initial program 41.3%

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

   3. Taylor expanded in c around inf 65.3%

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

      1. +-commutative 65.3%

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

      2. mul-1-neg 65.3%

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

      3. unsub-neg 65.3%

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

      4. associate-/l* 65.2%

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

      5. associate-/r/ 66.5%

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

   5. Simplified 66.5%

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

      1. *-un-lft-identity 66.5%

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

      2. pow2 66.5%

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

      3. times-frac 72.7%

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

   7. Applied egg-rr 72.7%

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

      1. associate-*l/ 72.7%

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

      2. *-lft-identity 72.7%

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

   9. Simplified 72.7%

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

       1. associate-*l/ 75.5%

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

   11. Applied egg-rr 75.5%

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

       1. associate-/l* 75.6%

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

   13. Simplified 75.6%

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

   if -1.84999999999999994e-34 < c < 5.7e-95 or 9.99999999999999953e-45 < c < 4.4e21

   1. Initial program 70.9%

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

   3. Taylor expanded in c around 0 74.5%

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

      1. associate-*r/ 74.5%

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

      2. neg-mul-1 74.5%

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

   5. Simplified 74.5%

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

   if 5.7e-95 < c < 9.99999999999999953e-45

   1. Initial program 89.5%

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

   3. Taylor expanded in c around inf 90.1%

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

      1. +-commutative 90.1%

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

      2. mul-1-neg 90.1%

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

      3. unsub-neg 90.1%

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

      4. associate-/l* 90.0%

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

      5. associate-/r/ 90.1%

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

   5. Simplified 90.1%

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

      1. *-un-lft-identity 90.1%

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

      2. pow2 90.1%

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

      3. times-frac 90.3%

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

   7. Applied egg-rr 90.3%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.85 \cdot 10^{-34}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{c}}{\frac{c}{d}}\\ \mathbf{elif}\;c \leq 5.7 \cdot 10^{-95}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 10^{-44}:\\ \;\;\;\;\frac{b}{c} + d \cdot \left(\frac{a}{c} \cdot \frac{-1}{c}\right)\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{+21}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{c}}{\frac{c}{d}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.2 \cdot 10^{-35} \lor \neg \left(c \leq 2.2 \cdot 10^{-92}\right) \land \left(c \leq 9.8 \cdot 10^{-45} \lor \neg \left(c \leq 4.4 \cdot 10^{+21}\right)\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -5.2e-35)
         (and (not (<= c 2.2e-92)) (or (<= c 9.8e-45) (not (<= c 4.4e+21)))))
   (/ b c)
   (/ (- a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.2e-35) || (!(c <= 2.2e-92) && ((c <= 9.8e-45) || !(c <= 4.4e+21)))) {
		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 <= (-5.2d-35)) .or. (.not. (c <= 2.2d-92)) .and. (c <= 9.8d-45) .or. (.not. (c <= 4.4d+21))) 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 <= -5.2e-35) || (!(c <= 2.2e-92) && ((c <= 9.8e-45) || !(c <= 4.4e+21)))) {
		tmp = b / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -5.2e-35) or (not (c <= 2.2e-92) and ((c <= 9.8e-45) or not (c <= 4.4e+21))):
		tmp = b / c
	else:
		tmp = -a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -5.2e-35) || (!(c <= 2.2e-92) && ((c <= 9.8e-45) || !(c <= 4.4e+21))))
		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 <= -5.2e-35) || (~((c <= 2.2e-92)) && ((c <= 9.8e-45) || ~((c <= 4.4e+21)))))
		tmp = b / c;
	else
		tmp = -a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -5.2e-35], And[N[Not[LessEqual[c, 2.2e-92]], $MachinePrecision], Or[LessEqual[c, 9.8e-45], N[Not[LessEqual[c, 4.4e+21]], $MachinePrecision]]]], N[(b / c), $MachinePrecision], N[((-a) / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -5.20000000000000009e-35 or 2.19999999999999987e-92 < c < 9.7999999999999996e-45 or 4.4e21 < c

   1. Initial program 44.5%

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

   3. Taylor expanded in c around inf 62.4%

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

   if -5.20000000000000009e-35 < c < 2.19999999999999987e-92 or 9.7999999999999996e-45 < c < 4.4e21

   1. Initial program 70.9%

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

   3. Taylor expanded in c around 0 74.5%

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

      1. associate-*r/ 74.5%

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

      2. neg-mul-1 74.5%

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

   5. Simplified 74.5%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.2 \cdot 10^{-35} \lor \neg \left(c \leq 2.2 \cdot 10^{-92}\right) \land \left(c \leq 9.8 \cdot 10^{-45} \lor \neg \left(c \leq 4.4 \cdot 10^{+21}\right)\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.2% 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 55.7%

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

3. Taylor expanded in c around inf 42.6%

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

4. Final simplification 42.6%

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

Developer target: 99.3% accurate, 0.1× speedup

Math

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

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 2024020 
(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))))