Complex division, imag part

Percentage Accurate: 61.7% → 95.6%

Time: 11.8s

Alternatives: 13

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: 61.7% 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: 95.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot c - a \cdot d\\ \mathbf{if}\;\frac{t_0}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{\mathsf{hypot}\left(c, d\right) \cdot \frac{\mathsf{hypot}\left(c, d\right)}{d}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* a d))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 5e+143)
     (* (/ 1.0 (hypot c d)) (/ t_0 (hypot c d)))
     (fma
      (/ c (hypot c d))
      (/ b (hypot c d))
      (/ (- a) (* (hypot c d) (/ (hypot c d) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (a * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 5e+143) {
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	} else {
		tmp = fma((c / hypot(c, d)), (b / hypot(c, d)), (-a / (hypot(c, d) * (hypot(c, d) / d))));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) - Float64(a * d))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= 5e+143)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, d)));
	else
		tmp = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(Float64(-a) / Float64(hypot(c, d) * Float64(hypot(c, d) / d))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+143], 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[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[((-a) / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] * N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / d), $MachinePrecision]), $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))) < 5.00000000000000012e143

   1. Initial program 80.1%

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

      1. *-un-lft-identity 80.1%

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

      2. add-sqr-sqrt 80.1%

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

      3. times-frac 80.2%

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

      4. hypot-def 80.2%

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

      5. hypot-def 98.0%

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

   3. Applied egg-rr 98.0%

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

   if 5.00000000000000012e143 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 24.6%

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

      1. div-sub 21.7%

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

      2. sub-neg 21.7%

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

      3. *-commutative 21.7%

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

      4. add-sqr-sqrt 21.7%

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

      5. times-frac 26.3%

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

      6. fma-def 26.3%

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

      7. hypot-def 26.3%

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

      8. hypot-def 69.5%

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

      9. associate-/l* 72.0%

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

      10. add-sqr-sqrt 72.0%

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

      11. pow2 72.0%

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

      12. hypot-def 72.0%

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

   3. Applied egg-rr 72.0%

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

      1. unpow2 72.0%

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

      2. *-un-lft-identity 72.0%

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

      3. times-frac 97.1%

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

   5. Applied egg-rr 97.1%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{\mathsf{hypot}\left(c, d\right) \cdot \frac{\mathsf{hypot}\left(c, d\right)}{d}}\right)\\ \end{array} \]

Alternative 2: 86.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot c - a \cdot d\\ \mathbf{if}\;\frac{t_0}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+271}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{\mathsf{hypot}\left(c, d\right) \cdot \frac{\mathsf{hypot}\left(c, d\right)}{d}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* a d))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 5e+271)
     (* (/ 1.0 (hypot c d)) (/ t_0 (hypot c d)))
     (fma 1.0 (/ b (hypot c d)) (/ (- a) (* (hypot c d) (/ (hypot c d) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (a * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 5e+271) {
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	} else {
		tmp = fma(1.0, (b / hypot(c, d)), (-a / (hypot(c, d) * (hypot(c, d) / d))));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) - Float64(a * d))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= 5e+271)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, d)));
	else
		tmp = fma(1.0, Float64(b / hypot(c, d)), Float64(Float64(-a) / Float64(hypot(c, d) * Float64(hypot(c, d) / d))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+271], 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[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[((-a) / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] * N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / d), $MachinePrecision]), $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))) < 5.0000000000000003e271

   1. Initial program 81.2%

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

      1. *-un-lft-identity 81.2%

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

      2. add-sqr-sqrt 81.2%

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

      3. times-frac 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 98.1%

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

   3. Applied egg-rr 98.1%

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

   if 5.0000000000000003e271 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 10.5%

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

      1. div-sub 7.0%

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

      2. sub-neg 7.0%

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

      3. *-commutative 7.0%

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

      4. add-sqr-sqrt 7.0%

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

      5. times-frac 12.4%

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

      6. fma-def 12.4%

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

      7. hypot-def 12.5%

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

      8. hypot-def 63.8%

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

      9. associate-/l* 66.8%

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

      10. add-sqr-sqrt 66.8%

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

      11. pow2 66.8%

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

      12. hypot-def 66.8%

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

   3. Applied egg-rr 66.8%

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

      1. unpow2 66.8%

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

      2. *-un-lft-identity 66.8%

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

      3. times-frac 96.6%

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

   5. Applied egg-rr 96.6%

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

   6. Taylor expanded in c around inf 72.5%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+271}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{\mathsf{hypot}\left(c, d\right) \cdot \frac{\mathsf{hypot}\left(c, d\right)}{d}}\right)\\ \end{array} \]

Alternative 3: 88.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot c - a \cdot d\\ \mathbf{if}\;\frac{t_0}{c \cdot c + d \cdot d} \leq 10^{+261}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{a}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* a d))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 1e+261)
     (* (/ 1.0 (hypot c d)) (/ t_0 (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 = (b * c) - (a * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+261) {
		tmp = (1.0 / hypot(c, d)) * (t_0 / 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(Float64(b * c) - Float64(a * d))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= 1e+261)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / 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[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+261], 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[(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 2 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 9.9999999999999993e260

   1. Initial program 81.1%

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

      1. *-un-lft-identity 81.1%

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

      2. add-sqr-sqrt 81.1%

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

      3. times-frac 81.2%

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

      4. hypot-def 81.2%

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

      5. hypot-def 98.1%

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

   3. Applied egg-rr 98.1%

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

   if 9.9999999999999993e260 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 12.0%

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

      1. div-sub 8.6%

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

      2. sub-neg 8.6%

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

      3. *-commutative 8.6%

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

      4. add-sqr-sqrt 8.6%

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

      5. times-frac 13.9%

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

      6. fma-def 13.9%

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

      7. hypot-def 13.9%

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

      8. hypot-def 64.4%

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

      9. associate-/l* 67.4%

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

      10. add-sqr-sqrt 67.4%

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

      11. pow2 67.4%

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

      12. hypot-def 67.4%

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

   3. Applied egg-rr 67.4%

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

   4. Taylor expanded in c around 0 70.3%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 10^{+261}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{a}{d}\right)\\ \end{array} \]

Alternative 4: 84.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot c - a \cdot d\\ \mathbf{if}\;\frac{t_0}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+248}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* a d))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 5e+248)
     (* (/ 1.0 (hypot c d)) (/ t_0 (hypot c d)))
     (- (/ b c) (/ (* a (/ d c)) c)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (a * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 5e+248) {
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	} else {
		tmp = (b / c) - ((a * (d / c)) / c);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (a * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 5e+248) {
		tmp = (1.0 / Math.hypot(c, d)) * (t_0 / Math.hypot(c, d));
	} else {
		tmp = (b / c) - ((a * (d / c)) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b * c) - (a * d)
	tmp = 0
	if (t_0 / ((c * c) + (d * d))) <= 5e+248:
		tmp = (1.0 / math.hypot(c, d)) * (t_0 / math.hypot(c, d))
	else:
		tmp = (b / c) - ((a * (d / c)) / c)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) - Float64(a * d))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= 5e+248)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, d)));
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(a * Float64(d / c)) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b * c) - (a * d);
	tmp = 0.0;
	if ((t_0 / ((c * c) + (d * d))) <= 5e+248)
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	else
		tmp = (b / c) - ((a * (d / c)) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+248], 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[(b / c), $MachinePrecision] - N[(N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision] / 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))) < 4.9999999999999996e248

   1. Initial program 81.0%

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

      1. *-un-lft-identity 81.0%

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

      2. add-sqr-sqrt 81.0%

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

      3. times-frac 81.1%

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

      4. hypot-def 81.1%

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

      5. hypot-def 98.1%

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

   3. Applied egg-rr 98.1%

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

   if 4.9999999999999996e248 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 13.4%

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

   2. Taylor expanded in c around inf 61.8%

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

      1. +-commutative 61.8%

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

      2. mul-1-neg 61.8%

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

      3. unsub-neg 61.8%

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

      4. *-commutative 61.8%

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

      5. associate-/l* 60.7%

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

   4. Simplified 60.7%

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

      1. *-un-lft-identity 60.7%

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

      2. add-sqr-sqrt 36.0%

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

      3. times-frac 36.0%

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

      4. sqrt-div 36.0%

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

      5. unpow2 36.0%

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

      6. sqrt-prod 25.4%

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

      7. add-sqr-sqrt 34.1%

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

      8. sqrt-div 34.1%

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

      9. unpow2 34.1%

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

      10. sqrt-prod 27.0%

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

      11. add-sqr-sqrt 43.7%

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

   6. Applied egg-rr 43.7%

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

      1. associate-*l/ 43.7%

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

      2. *-lft-identity 43.7%

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

      3. associate-/r/ 43.7%

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

   8. Simplified 43.7%

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

      1. sub-neg 43.7%

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

      2. associate-/r/ 42.2%

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

   10. Applied egg-rr 42.2%

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

       1. sub-neg 42.2%

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

       2. associate-*l/ 43.8%

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

       3. associate-*l* 43.8%

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

       4. rem-square-sqrt 68.5%

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

   12. Simplified 68.5%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+248}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 5: 79.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \mathbf{if}\;c \leq -4 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -1.8 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 3.3 \cdot 10^{-102}:\\ \;\;\;\;\frac{1}{\frac{{d}^{2}}{b \cdot c}} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{+142}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))
        (t_1 (- (/ b c) (/ (* a (/ d c)) c))))
   (if (<= c -4e+123)
     t_1
     (if (<= c -1.8e-105)
       t_0
       (if (<= c 3.3e-102)
         (- (/ 1.0 (/ (pow d 2.0) (* b c))) (/ a d))
         (if (<= c 1.7e+142) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = (b / c) - ((a * (d / c)) / c);
	double tmp;
	if (c <= -4e+123) {
		tmp = t_1;
	} else if (c <= -1.8e-105) {
		tmp = t_0;
	} else if (c <= 3.3e-102) {
		tmp = (1.0 / (pow(d, 2.0) / (b * c))) - (a / d);
	} else if (c <= 1.7e+142) {
		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 = ((b * c) - (a * d)) / ((c * c) + (d * d))
    t_1 = (b / c) - ((a * (d / c)) / c)
    if (c <= (-4d+123)) then
        tmp = t_1
    else if (c <= (-1.8d-105)) then
        tmp = t_0
    else if (c <= 3.3d-102) then
        tmp = (1.0d0 / ((d ** 2.0d0) / (b * c))) - (a / d)
    else if (c <= 1.7d+142) 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 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = (b / c) - ((a * (d / c)) / c);
	double tmp;
	if (c <= -4e+123) {
		tmp = t_1;
	} else if (c <= -1.8e-105) {
		tmp = t_0;
	} else if (c <= 3.3e-102) {
		tmp = (1.0 / (Math.pow(d, 2.0) / (b * c))) - (a / d);
	} else if (c <= 1.7e+142) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
	t_1 = (b / c) - ((a * (d / c)) / c)
	tmp = 0
	if c <= -4e+123:
		tmp = t_1
	elif c <= -1.8e-105:
		tmp = t_0
	elif c <= 3.3e-102:
		tmp = (1.0 / (math.pow(d, 2.0) / (b * c))) - (a / d)
	elif c <= 1.7e+142:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b / c) - Float64(Float64(a * Float64(d / c)) / c))
	tmp = 0.0
	if (c <= -4e+123)
		tmp = t_1;
	elseif (c <= -1.8e-105)
		tmp = t_0;
	elseif (c <= 3.3e-102)
		tmp = Float64(Float64(1.0 / Float64((d ^ 2.0) / Float64(b * c))) - Float64(a / d));
	elseif (c <= 1.7e+142)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	t_1 = (b / c) - ((a * (d / c)) / c);
	tmp = 0.0;
	if (c <= -4e+123)
		tmp = t_1;
	elseif (c <= -1.8e-105)
		tmp = t_0;
	elseif (c <= 3.3e-102)
		tmp = (1.0 / ((d ^ 2.0) / (b * c))) - (a / d);
	elseif (c <= 1.7e+142)
		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[(b * c), $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 * N[(d / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -4e+123], t$95$1, If[LessEqual[c, -1.8e-105], t$95$0, If[LessEqual[c, 3.3e-102], N[(N[(1.0 / N[(N[Power[d, 2.0], $MachinePrecision] / N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.7e+142], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.99999999999999991e123 or 1.6999999999999999e142 < c

   1. Initial program 33.2%

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

   2. Taylor expanded in c around inf 86.9%

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

      1. +-commutative 86.9%

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

      2. mul-1-neg 86.9%

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

      3. unsub-neg 86.9%

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

      4. *-commutative 86.9%

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

      5. associate-/l* 87.2%

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

   4. Simplified 87.2%

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

      1. *-un-lft-identity 87.2%

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

      2. add-sqr-sqrt 47.3%

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

      3. times-frac 47.3%

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

      4. sqrt-div 47.3%

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

      5. unpow2 47.3%

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

      6. sqrt-prod 28.8%

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

      7. add-sqr-sqrt 47.3%

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

      8. sqrt-div 47.3%

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

      9. unpow2 47.3%

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

      10. sqrt-prod 31.2%

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

      11. add-sqr-sqrt 54.5%

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

   6. Applied egg-rr 54.5%

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

      1. associate-*l/ 54.5%

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

      2. *-lft-identity 54.5%

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

      3. associate-/r/ 54.6%

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

   8. Simplified 54.6%

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

      1. sub-neg 54.6%

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

      2. associate-/r/ 52.2%

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

   10. Applied egg-rr 52.2%

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

       1. sub-neg 52.2%

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

       2. associate-*l/ 54.6%

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

       3. associate-*l* 54.6%

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

       4. rem-square-sqrt 94.6%

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

   12. Simplified 94.6%

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

   if -3.99999999999999991e123 < c < -1.79999999999999982e-105 or 3.3e-102 < c < 1.6999999999999999e142

   1. Initial program 85.2%

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

   if -1.79999999999999982e-105 < c < 3.3e-102

   1. Initial program 71.8%

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

   2. Taylor expanded in c around 0 84.7%

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

      1. +-commutative 84.7%

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

      2. mul-1-neg 84.7%

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

      3. unsub-neg 84.7%

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

      4. associate-/l* 88.4%

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

      5. associate-/r/ 80.5%

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

   4. Simplified 80.5%

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

      1. pow2 80.5%

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

      2. associate-*l/ 84.7%

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

      3. clear-num 84.7%

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

      4. pow2 84.7%

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

      5. *-commutative 84.7%

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

   6. Applied egg-rr 84.7%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4 \cdot 10^{+123}:\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -1.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 3.3 \cdot 10^{-102}:\\ \;\;\;\;\frac{1}{\frac{{d}^{2}}{b \cdot c}} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{+142}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 6: 79.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \mathbf{if}\;c \leq -8 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -3.8 \cdot 10^{-147}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{-103}:\\ \;\;\;\;\left(b \cdot c\right) \cdot {d}^{-2} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{+142}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))
        (t_1 (- (/ b c) (/ (* a (/ d c)) c))))
   (if (<= c -8e+122)
     t_1
     (if (<= c -3.8e-147)
       t_0
       (if (<= c 5.2e-103)
         (- (* (* b c) (pow d -2.0)) (/ a d))
         (if (<= c 1.7e+142) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = (b / c) - ((a * (d / c)) / c);
	double tmp;
	if (c <= -8e+122) {
		tmp = t_1;
	} else if (c <= -3.8e-147) {
		tmp = t_0;
	} else if (c <= 5.2e-103) {
		tmp = ((b * c) * pow(d, -2.0)) - (a / d);
	} else if (c <= 1.7e+142) {
		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 = ((b * c) - (a * d)) / ((c * c) + (d * d))
    t_1 = (b / c) - ((a * (d / c)) / c)
    if (c <= (-8d+122)) then
        tmp = t_1
    else if (c <= (-3.8d-147)) then
        tmp = t_0
    else if (c <= 5.2d-103) then
        tmp = ((b * c) * (d ** (-2.0d0))) - (a / d)
    else if (c <= 1.7d+142) 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 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = (b / c) - ((a * (d / c)) / c);
	double tmp;
	if (c <= -8e+122) {
		tmp = t_1;
	} else if (c <= -3.8e-147) {
		tmp = t_0;
	} else if (c <= 5.2e-103) {
		tmp = ((b * c) * Math.pow(d, -2.0)) - (a / d);
	} else if (c <= 1.7e+142) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
	t_1 = (b / c) - ((a * (d / c)) / c)
	tmp = 0
	if c <= -8e+122:
		tmp = t_1
	elif c <= -3.8e-147:
		tmp = t_0
	elif c <= 5.2e-103:
		tmp = ((b * c) * math.pow(d, -2.0)) - (a / d)
	elif c <= 1.7e+142:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b / c) - Float64(Float64(a * Float64(d / c)) / c))
	tmp = 0.0
	if (c <= -8e+122)
		tmp = t_1;
	elseif (c <= -3.8e-147)
		tmp = t_0;
	elseif (c <= 5.2e-103)
		tmp = Float64(Float64(Float64(b * c) * (d ^ -2.0)) - Float64(a / d));
	elseif (c <= 1.7e+142)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	t_1 = (b / c) - ((a * (d / c)) / c);
	tmp = 0.0;
	if (c <= -8e+122)
		tmp = t_1;
	elseif (c <= -3.8e-147)
		tmp = t_0;
	elseif (c <= 5.2e-103)
		tmp = ((b * c) * (d ^ -2.0)) - (a / d);
	elseif (c <= 1.7e+142)
		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[(b * c), $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 * N[(d / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -8e+122], t$95$1, If[LessEqual[c, -3.8e-147], t$95$0, If[LessEqual[c, 5.2e-103], N[(N[(N[(b * c), $MachinePrecision] * N[Power[d, -2.0], $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.7e+142], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -8.00000000000000012e122 or 1.6999999999999999e142 < c

   1. Initial program 33.2%

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

   2. Taylor expanded in c around inf 86.9%

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

      1. +-commutative 86.9%

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

      2. mul-1-neg 86.9%

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

      3. unsub-neg 86.9%

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

      4. *-commutative 86.9%

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

      5. associate-/l* 87.2%

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

   4. Simplified 87.2%

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

      1. *-un-lft-identity 87.2%

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

      2. add-sqr-sqrt 47.3%

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

      3. times-frac 47.3%

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

      4. sqrt-div 47.3%

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

      5. unpow2 47.3%

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

      6. sqrt-prod 28.8%

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

      7. add-sqr-sqrt 47.3%

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

      8. sqrt-div 47.3%

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

      9. unpow2 47.3%

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

      10. sqrt-prod 31.2%

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

      11. add-sqr-sqrt 54.5%

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

   6. Applied egg-rr 54.5%

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

      1. associate-*l/ 54.5%

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

      2. *-lft-identity 54.5%

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

      3. associate-/r/ 54.6%

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

   8. Simplified 54.6%

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

      1. sub-neg 54.6%

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

      2. associate-/r/ 52.2%

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

   10. Applied egg-rr 52.2%

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

       1. sub-neg 52.2%

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

       2. associate-*l/ 54.6%

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

       3. associate-*l* 54.6%

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

       4. rem-square-sqrt 94.6%

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

   12. Simplified 94.6%

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

   if -8.00000000000000012e122 < c < -3.80000000000000028e-147 or 5.19999999999999993e-103 < c < 1.6999999999999999e142

   1. Initial program 85.5%

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

   if -3.80000000000000028e-147 < c < 5.19999999999999993e-103

   1. Initial program 69.8%

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

   2. Taylor expanded in c around 0 84.2%

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

      1. +-commutative 84.2%

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

      2. mul-1-neg 84.2%

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

      3. unsub-neg 84.2%

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

      4. associate-/l* 88.4%

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

      5. associate-/r/ 80.8%

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

   4. Simplified 80.8%

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

      1. add-cbrt-cube 57.7%

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

      2. pow1/3 32.2%

         \[\leadsto \color{blue}{{\left(\left(\left(\frac{b}{{d}^{2}} \cdot c - \frac{a}{d}\right) \cdot \left(\frac{b}{{d}^{2}} \cdot c - \frac{a}{d}\right)\right) \cdot \left(\frac{b}{{d}^{2}} \cdot c - \frac{a}{d}\right)\right)}^{0.3333333333333333}} \]

      3. pow3 32.2%

         \[\leadsto {\color{blue}{\left({\left(\frac{b}{{d}^{2}} \cdot c - \frac{a}{d}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. *-commutative 32.2%

         \[\leadsto {\left({\left(\color{blue}{c \cdot \frac{b}{{d}^{2}}} - \frac{a}{d}\right)}^{3}\right)}^{0.3333333333333333} \]

      5. pow2 32.2%

         \[\leadsto {\left({\left(c \cdot \frac{b}{\color{blue}{d \cdot d}} - \frac{a}{d}\right)}^{3}\right)}^{0.3333333333333333} \]

      6. fma-neg 32.2%

         \[\leadsto {\left({\color{blue}{\left(\mathsf{fma}\left(c, \frac{b}{d \cdot d}, -\frac{a}{d}\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

      7. div-inv 32.2%

         \[\leadsto {\left({\left(\mathsf{fma}\left(c, \color{blue}{b \cdot \frac{1}{d \cdot d}}, -\frac{a}{d}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

      8. pow2 32.2%

         \[\leadsto {\left({\left(\mathsf{fma}\left(c, b \cdot \frac{1}{\color{blue}{{d}^{2}}}, -\frac{a}{d}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

      9. pow-flip 32.2%

         \[\leadsto {\left({\left(\mathsf{fma}\left(c, b \cdot \color{blue}{{d}^{\left(-2\right)}}, -\frac{a}{d}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

      10. metadata-eval 32.2%

          \[\leadsto {\left({\left(\mathsf{fma}\left(c, b \cdot {d}^{\color{blue}{-2}}, -\frac{a}{d}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 32.2%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(c, b \cdot {d}^{-2}, -\frac{a}{d}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
   7. Step-by-step derivation

      1. unpow1/3 57.7%

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

      2. rem-cbrt-cube 80.8%

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

      3. fma-udef 80.8%

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

      4. unsub-neg 80.8%

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

   8. Applied egg-rr 80.8%

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

      1. associate-*r* 84.2%

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

   10. Simplified 84.2%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8 \cdot 10^{+122}:\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -3.8 \cdot 10^{-147}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{-103}:\\ \;\;\;\;\left(b \cdot c\right) \cdot {d}^{-2} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{+142}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 7: 79.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \mathbf{if}\;c \leq -3.4 \cdot 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -7.5 \cdot 10^{-147}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{-86}:\\ \;\;\;\;c \cdot \left(\frac{1}{d} \cdot \frac{b}{d}\right) - \frac{a}{d}\\ \mathbf{elif}\;c \leq 2.35 \cdot 10^{+142}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))
        (t_1 (- (/ b c) (/ (* a (/ d c)) c))))
   (if (<= c -3.4e+120)
     t_1
     (if (<= c -7.5e-147)
       t_0
       (if (<= c 7.2e-86)
         (- (* c (* (/ 1.0 d) (/ b d))) (/ a d))
         (if (<= c 2.35e+142) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = (b / c) - ((a * (d / c)) / c);
	double tmp;
	if (c <= -3.4e+120) {
		tmp = t_1;
	} else if (c <= -7.5e-147) {
		tmp = t_0;
	} else if (c <= 7.2e-86) {
		tmp = (c * ((1.0 / d) * (b / d))) - (a / d);
	} else if (c <= 2.35e+142) {
		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 = ((b * c) - (a * d)) / ((c * c) + (d * d))
    t_1 = (b / c) - ((a * (d / c)) / c)
    if (c <= (-3.4d+120)) then
        tmp = t_1
    else if (c <= (-7.5d-147)) then
        tmp = t_0
    else if (c <= 7.2d-86) then
        tmp = (c * ((1.0d0 / d) * (b / d))) - (a / d)
    else if (c <= 2.35d+142) 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 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = (b / c) - ((a * (d / c)) / c);
	double tmp;
	if (c <= -3.4e+120) {
		tmp = t_1;
	} else if (c <= -7.5e-147) {
		tmp = t_0;
	} else if (c <= 7.2e-86) {
		tmp = (c * ((1.0 / d) * (b / d))) - (a / d);
	} else if (c <= 2.35e+142) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
	t_1 = (b / c) - ((a * (d / c)) / c)
	tmp = 0
	if c <= -3.4e+120:
		tmp = t_1
	elif c <= -7.5e-147:
		tmp = t_0
	elif c <= 7.2e-86:
		tmp = (c * ((1.0 / d) * (b / d))) - (a / d)
	elif c <= 2.35e+142:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b / c) - Float64(Float64(a * Float64(d / c)) / c))
	tmp = 0.0
	if (c <= -3.4e+120)
		tmp = t_1;
	elseif (c <= -7.5e-147)
		tmp = t_0;
	elseif (c <= 7.2e-86)
		tmp = Float64(Float64(c * Float64(Float64(1.0 / d) * Float64(b / d))) - Float64(a / d));
	elseif (c <= 2.35e+142)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	t_1 = (b / c) - ((a * (d / c)) / c);
	tmp = 0.0;
	if (c <= -3.4e+120)
		tmp = t_1;
	elseif (c <= -7.5e-147)
		tmp = t_0;
	elseif (c <= 7.2e-86)
		tmp = (c * ((1.0 / d) * (b / d))) - (a / d);
	elseif (c <= 2.35e+142)
		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[(b * c), $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 * N[(d / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3.4e+120], t$95$1, If[LessEqual[c, -7.5e-147], t$95$0, If[LessEqual[c, 7.2e-86], N[(N[(c * N[(N[(1.0 / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.35e+142], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.39999999999999999e120 or 2.35e142 < c

   1. Initial program 33.2%

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

   2. Taylor expanded in c around inf 86.9%

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

      1. +-commutative 86.9%

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

      2. mul-1-neg 86.9%

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

      3. unsub-neg 86.9%

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

      4. *-commutative 86.9%

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

      5. associate-/l* 87.2%

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

   4. Simplified 87.2%

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

      1. *-un-lft-identity 87.2%

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

      2. add-sqr-sqrt 47.3%

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

      3. times-frac 47.3%

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

      4. sqrt-div 47.3%

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

      5. unpow2 47.3%

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

      6. sqrt-prod 28.8%

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

      7. add-sqr-sqrt 47.3%

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

      8. sqrt-div 47.3%

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

      9. unpow2 47.3%

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

      10. sqrt-prod 31.2%

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

      11. add-sqr-sqrt 54.5%

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

   6. Applied egg-rr 54.5%

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

      1. associate-*l/ 54.5%

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

      2. *-lft-identity 54.5%

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

      3. associate-/r/ 54.6%

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

   8. Simplified 54.6%

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

      1. sub-neg 54.6%

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

      2. associate-/r/ 52.2%

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

   10. Applied egg-rr 52.2%

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

       1. sub-neg 52.2%

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

       2. associate-*l/ 54.6%

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

       3. associate-*l* 54.6%

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

       4. rem-square-sqrt 94.6%

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

   12. Simplified 94.6%

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

   if -3.39999999999999999e120 < c < -7.50000000000000047e-147 or 7.19999999999999932e-86 < c < 2.35e142

   1. Initial program 86.1%

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

   if -7.50000000000000047e-147 < c < 7.19999999999999932e-86

   1. Initial program 69.7%

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

   2. Taylor expanded in c around 0 82.3%

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

      1. +-commutative 82.3%

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

      2. mul-1-neg 82.3%

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

      3. unsub-neg 82.3%

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

      4. associate-/l* 86.3%

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

      5. associate-/r/ 79.0%

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

   4. Simplified 79.0%

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

      1. *-un-lft-identity 79.0%

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

      2. pow2 79.0%

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

      3. times-frac 82.8%

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

   6. Applied egg-rr 82.8%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.4 \cdot 10^{+120}:\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -7.5 \cdot 10^{-147}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{-86}:\\ \;\;\;\;c \cdot \left(\frac{1}{d} \cdot \frac{b}{d}\right) - \frac{a}{d}\\ \mathbf{elif}\;c \leq 2.35 \cdot 10^{+142}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 8: 73.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.6 \cdot 10^{-9} \lor \neg \left(c \leq 3.8 \cdot 10^{-63} \lor \neg \left(c \leq 240000000000\right) \land c \leq 7.2 \cdot 10^{+47}\right):\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{1}{d} \cdot \frac{b}{d}\right) - \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.6e-9)
         (not
          (or (<= c 3.8e-63)
              (and (not (<= c 240000000000.0)) (<= c 7.2e+47)))))
   (- (/ b c) (/ (* a (/ d c)) c))
   (- (* c (* (/ 1.0 d) (/ b d))) (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.6e-9) || !((c <= 3.8e-63) || (!(c <= 240000000000.0) && (c <= 7.2e+47)))) {
		tmp = (b / c) - ((a * (d / c)) / c);
	} else {
		tmp = (c * ((1.0 / d) * (b / d))) - (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 <= (-1.6d-9)) .or. (.not. (c <= 3.8d-63) .or. (.not. (c <= 240000000000.0d0)) .and. (c <= 7.2d+47))) then
        tmp = (b / c) - ((a * (d / c)) / c)
    else
        tmp = (c * ((1.0d0 / d) * (b / d))) - (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 <= -1.6e-9) || !((c <= 3.8e-63) || (!(c <= 240000000000.0) && (c <= 7.2e+47)))) {
		tmp = (b / c) - ((a * (d / c)) / c);
	} else {
		tmp = (c * ((1.0 / d) * (b / d))) - (a / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.6e-9) or not ((c <= 3.8e-63) or (not (c <= 240000000000.0) and (c <= 7.2e+47))):
		tmp = (b / c) - ((a * (d / c)) / c)
	else:
		tmp = (c * ((1.0 / d) * (b / d))) - (a / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.6e-9) || !((c <= 3.8e-63) || (!(c <= 240000000000.0) && (c <= 7.2e+47))))
		tmp = Float64(Float64(b / c) - Float64(Float64(a * Float64(d / c)) / c));
	else
		tmp = Float64(Float64(c * Float64(Float64(1.0 / d) * Float64(b / d))) - Float64(a / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.6e-9) || ~(((c <= 3.8e-63) || (~((c <= 240000000000.0)) && (c <= 7.2e+47)))))
		tmp = (b / c) - ((a * (d / c)) / c);
	else
		tmp = (c * ((1.0 / d) * (b / d))) - (a / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.6e-9], N[Not[Or[LessEqual[c, 3.8e-63], And[N[Not[LessEqual[c, 240000000000.0]], $MachinePrecision], LessEqual[c, 7.2e+47]]]], $MachinePrecision]], N[(N[(b / c), $MachinePrecision] - N[(N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(1.0 / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.60000000000000006e-9 or 3.80000000000000017e-63 < c < 2.4e11 or 7.20000000000000015e47 < c

   1. Initial program 59.0%

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

   2. Taylor expanded in c around inf 83.9%

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

      1. +-commutative 83.9%

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

      2. mul-1-neg 83.9%

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

      3. unsub-neg 83.9%

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

      4. *-commutative 83.9%

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

      5. associate-/l* 83.3%

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

   4. Simplified 83.3%

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

      1. *-un-lft-identity 83.3%

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

      2. add-sqr-sqrt 44.6%

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

      3. times-frac 44.6%

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

      4. sqrt-div 44.7%

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

      5. unpow2 44.7%

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

      6. sqrt-prod 26.3%

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

      7. add-sqr-sqrt 43.2%

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

      8. sqrt-div 44.0%

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

      9. unpow2 44.0%

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

      10. sqrt-prod 28.4%

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

      11. add-sqr-sqrt 49.5%

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

   6. Applied egg-rr 49.5%

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

      1. associate-*l/ 49.5%

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

      2. *-lft-identity 49.5%

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

      3. associate-/r/ 49.0%

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

   8. Simplified 49.0%

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

      1. sub-neg 49.0%

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

      2. associate-/r/ 47.7%

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

   10. Applied egg-rr 47.7%

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

       1. sub-neg 47.7%

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

       2. associate-*l/ 49.0%

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

       3. associate-*l* 49.0%

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

       4. rem-square-sqrt 85.8%

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

   12. Simplified 85.8%

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

   if -1.60000000000000006e-9 < c < 3.80000000000000017e-63 or 2.4e11 < c < 7.20000000000000015e47

   1. Initial program 72.4%

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

   2. Taylor expanded in c around 0 76.7%

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

      1. +-commutative 76.7%

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

      2. mul-1-neg 76.7%

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

      3. unsub-neg 76.7%

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

      4. associate-/l* 79.4%

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

      5. associate-/r/ 73.8%

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

   4. Simplified 73.8%

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

      1. *-un-lft-identity 73.8%

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

      2. pow2 73.8%

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

      3. times-frac 78.8%

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

   6. Applied egg-rr 78.8%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.6 \cdot 10^{-9} \lor \neg \left(c \leq 3.8 \cdot 10^{-63} \lor \neg \left(c \leq 240000000000\right) \land c \leq 7.2 \cdot 10^{+47}\right):\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{1}{d} \cdot \frac{b}{d}\right) - \frac{a}{d}\\ \end{array} \]

Alternative 9: 73.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(\frac{1}{d} \cdot \frac{b}{d}\right) - \frac{a}{d}\\ t_1 := a \cdot \frac{d}{c}\\ t_2 := \frac{b}{c} - \frac{t_1}{c}\\ \mathbf{if}\;c \leq -1.6 \cdot 10^{-10}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 265000000000:\\ \;\;\;\;\frac{b}{c} + \frac{-1}{\frac{c}{t_1}}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+47}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c (* (/ 1.0 d) (/ b d))) (/ a d)))
        (t_1 (* a (/ d c)))
        (t_2 (- (/ b c) (/ t_1 c))))
   (if (<= c -1.6e-10)
     t_2
     (if (<= c 3.8e-63)
       t_0
       (if (<= c 265000000000.0)
         (+ (/ b c) (/ -1.0 (/ c t_1)))
         (if (<= c 7.2e+47) t_0 t_2))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * ((1.0 / d) * (b / d))) - (a / d);
	double t_1 = a * (d / c);
	double t_2 = (b / c) - (t_1 / c);
	double tmp;
	if (c <= -1.6e-10) {
		tmp = t_2;
	} else if (c <= 3.8e-63) {
		tmp = t_0;
	} else if (c <= 265000000000.0) {
		tmp = (b / c) + (-1.0 / (c / t_1));
	} else if (c <= 7.2e+47) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (c * ((1.0d0 / d) * (b / d))) - (a / d)
    t_1 = a * (d / c)
    t_2 = (b / c) - (t_1 / c)
    if (c <= (-1.6d-10)) then
        tmp = t_2
    else if (c <= 3.8d-63) then
        tmp = t_0
    else if (c <= 265000000000.0d0) then
        tmp = (b / c) + ((-1.0d0) / (c / t_1))
    else if (c <= 7.2d+47) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (c * ((1.0 / d) * (b / d))) - (a / d);
	double t_1 = a * (d / c);
	double t_2 = (b / c) - (t_1 / c);
	double tmp;
	if (c <= -1.6e-10) {
		tmp = t_2;
	} else if (c <= 3.8e-63) {
		tmp = t_0;
	} else if (c <= 265000000000.0) {
		tmp = (b / c) + (-1.0 / (c / t_1));
	} else if (c <= 7.2e+47) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (c * ((1.0 / d) * (b / d))) - (a / d)
	t_1 = a * (d / c)
	t_2 = (b / c) - (t_1 / c)
	tmp = 0
	if c <= -1.6e-10:
		tmp = t_2
	elif c <= 3.8e-63:
		tmp = t_0
	elif c <= 265000000000.0:
		tmp = (b / c) + (-1.0 / (c / t_1))
	elif c <= 7.2e+47:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * Float64(Float64(1.0 / d) * Float64(b / d))) - Float64(a / d))
	t_1 = Float64(a * Float64(d / c))
	t_2 = Float64(Float64(b / c) - Float64(t_1 / c))
	tmp = 0.0
	if (c <= -1.6e-10)
		tmp = t_2;
	elseif (c <= 3.8e-63)
		tmp = t_0;
	elseif (c <= 265000000000.0)
		tmp = Float64(Float64(b / c) + Float64(-1.0 / Float64(c / t_1)));
	elseif (c <= 7.2e+47)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (c * ((1.0 / d) * (b / d))) - (a / d);
	t_1 = a * (d / c);
	t_2 = (b / c) - (t_1 / c);
	tmp = 0.0;
	if (c <= -1.6e-10)
		tmp = t_2;
	elseif (c <= 3.8e-63)
		tmp = t_0;
	elseif (c <= 265000000000.0)
		tmp = (b / c) + (-1.0 / (c / t_1));
	elseif (c <= 7.2e+47)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * N[(N[(1.0 / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b / c), $MachinePrecision] - N[(t$95$1 / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.6e-10], t$95$2, If[LessEqual[c, 3.8e-63], t$95$0, If[LessEqual[c, 265000000000.0], N[(N[(b / c), $MachinePrecision] + N[(-1.0 / N[(c / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 7.2e+47], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.5999999999999999e-10 or 7.20000000000000015e47 < c

   1. Initial program 53.8%

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

   2. Taylor expanded in c around inf 86.0%

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

      1. +-commutative 86.0%

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

      2. mul-1-neg 86.0%

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

      3. unsub-neg 86.0%

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

      4. *-commutative 86.0%

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

      5. associate-/l* 86.2%

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

   4. Simplified 86.2%

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

      1. *-un-lft-identity 86.2%

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

      2. add-sqr-sqrt 46.2%

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

      3. times-frac 46.2%

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

      4. sqrt-div 46.2%

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

      5. unpow2 46.2%

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

      6. sqrt-prod 25.1%

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

      7. add-sqr-sqrt 44.6%

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

      8. sqrt-div 44.6%

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

      9. unpow2 44.6%

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

      10. sqrt-prod 26.6%

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

      11. add-sqr-sqrt 50.9%

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

   6. Applied egg-rr 50.9%

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

      1. associate-*l/ 50.9%

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

      2. *-lft-identity 50.9%

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

      3. associate-/r/ 50.3%

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

   8. Simplified 50.3%

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

      1. sub-neg 50.3%

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

      2. associate-/r/ 48.8%

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

   10. Applied egg-rr 48.8%

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

       1. sub-neg 48.8%

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

       2. associate-*l/ 50.3%

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

       3. associate-*l* 50.3%

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

       4. rem-square-sqrt 88.2%

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

   12. Simplified 88.2%

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

   if -1.5999999999999999e-10 < c < 3.80000000000000017e-63 or 2.65e11 < c < 7.20000000000000015e47

   1. Initial program 72.4%

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

   2. Taylor expanded in c around 0 76.7%

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

      1. +-commutative 76.7%

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

      2. mul-1-neg 76.7%

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

      3. unsub-neg 76.7%

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

      4. associate-/l* 79.4%

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

      5. associate-/r/ 73.8%

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

   4. Simplified 73.8%

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

      1. *-un-lft-identity 73.8%

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

      2. pow2 73.8%

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

      3. times-frac 78.8%

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

   6. Applied egg-rr 78.8%

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

   if 3.80000000000000017e-63 < c < 2.65e11

   1. Initial program 94.1%

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

   2. Taylor expanded in c around inf 69.9%

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

      1. +-commutative 69.9%

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

      2. mul-1-neg 69.9%

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

      3. unsub-neg 69.9%

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

      4. *-commutative 69.9%

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

      5. associate-/l* 64.0%

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

   4. Simplified 64.0%

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

      1. *-un-lft-identity 64.0%

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

      2. add-sqr-sqrt 34.2%

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

      3. times-frac 34.2%

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

      4. sqrt-div 34.4%

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

      5. unpow2 34.4%

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

      6. sqrt-prod 34.4%

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

      7. add-sqr-sqrt 34.4%

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

      8. sqrt-div 40.1%

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

      9. unpow2 40.1%

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

      10. sqrt-prod 40.2%

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

      11. add-sqr-sqrt 40.1%

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

   6. Applied egg-rr 40.1%

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

      1. associate-*l/ 40.1%

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

      2. *-lft-identity 40.1%

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

      3. associate-/r/ 40.1%

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

   8. Simplified 40.1%

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

      1. clear-num 40.3%

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

      2. inv-pow 40.3%

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

   10. Applied egg-rr 40.3%

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

       1. unpow-1 40.3%

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

       2. associate-/l/ 40.4%

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

       3. associate-*l* 40.3%

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

       4. rem-square-sqrt 70.1%

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

   12. Simplified 70.1%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{-63}:\\ \;\;\;\;c \cdot \left(\frac{1}{d} \cdot \frac{b}{d}\right) - \frac{a}{d}\\ \mathbf{elif}\;c \leq 265000000000:\\ \;\;\;\;\frac{b}{c} + \frac{-1}{\frac{c}{a \cdot \frac{d}{c}}}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+47}:\\ \;\;\;\;c \cdot \left(\frac{1}{d} \cdot \frac{b}{d}\right) - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 10: 73.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(\frac{1}{d} \cdot \frac{b}{d}\right) - \frac{a}{d}\\ t_1 := a \cdot \frac{d}{c}\\ \mathbf{if}\;c \leq -4 \cdot 10^{-9}:\\ \;\;\;\;\frac{b}{c} - \frac{t_1}{c}\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 240000000000:\\ \;\;\;\;\frac{b}{c} + \frac{-1}{\frac{c}{t_1}}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+47}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c \cdot \frac{c}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c (* (/ 1.0 d) (/ b d))) (/ a d))) (t_1 (* a (/ d c))))
   (if (<= c -4e-9)
     (- (/ b c) (/ t_1 c))
     (if (<= c 3.8e-63)
       t_0
       (if (<= c 240000000000.0)
         (+ (/ b c) (/ -1.0 (/ c t_1)))
         (if (<= c 7.2e+47) t_0 (- (/ b c) (/ d (* c (/ c a))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * ((1.0 / d) * (b / d))) - (a / d);
	double t_1 = a * (d / c);
	double tmp;
	if (c <= -4e-9) {
		tmp = (b / c) - (t_1 / c);
	} else if (c <= 3.8e-63) {
		tmp = t_0;
	} else if (c <= 240000000000.0) {
		tmp = (b / c) + (-1.0 / (c / t_1));
	} else if (c <= 7.2e+47) {
		tmp = t_0;
	} else {
		tmp = (b / c) - (d / (c * (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (c * ((1.0d0 / d) * (b / d))) - (a / d)
    t_1 = a * (d / c)
    if (c <= (-4d-9)) then
        tmp = (b / c) - (t_1 / c)
    else if (c <= 3.8d-63) then
        tmp = t_0
    else if (c <= 240000000000.0d0) then
        tmp = (b / c) + ((-1.0d0) / (c / t_1))
    else if (c <= 7.2d+47) then
        tmp = t_0
    else
        tmp = (b / c) - (d / (c * (c / a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (c * ((1.0 / d) * (b / d))) - (a / d);
	double t_1 = a * (d / c);
	double tmp;
	if (c <= -4e-9) {
		tmp = (b / c) - (t_1 / c);
	} else if (c <= 3.8e-63) {
		tmp = t_0;
	} else if (c <= 240000000000.0) {
		tmp = (b / c) + (-1.0 / (c / t_1));
	} else if (c <= 7.2e+47) {
		tmp = t_0;
	} else {
		tmp = (b / c) - (d / (c * (c / a)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (c * ((1.0 / d) * (b / d))) - (a / d)
	t_1 = a * (d / c)
	tmp = 0
	if c <= -4e-9:
		tmp = (b / c) - (t_1 / c)
	elif c <= 3.8e-63:
		tmp = t_0
	elif c <= 240000000000.0:
		tmp = (b / c) + (-1.0 / (c / t_1))
	elif c <= 7.2e+47:
		tmp = t_0
	else:
		tmp = (b / c) - (d / (c * (c / a)))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * Float64(Float64(1.0 / d) * Float64(b / d))) - Float64(a / d))
	t_1 = Float64(a * Float64(d / c))
	tmp = 0.0
	if (c <= -4e-9)
		tmp = Float64(Float64(b / c) - Float64(t_1 / c));
	elseif (c <= 3.8e-63)
		tmp = t_0;
	elseif (c <= 240000000000.0)
		tmp = Float64(Float64(b / c) + Float64(-1.0 / Float64(c / t_1)));
	elseif (c <= 7.2e+47)
		tmp = t_0;
	else
		tmp = Float64(Float64(b / c) - Float64(d / Float64(c * Float64(c / a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (c * ((1.0 / d) * (b / d))) - (a / d);
	t_1 = a * (d / c);
	tmp = 0.0;
	if (c <= -4e-9)
		tmp = (b / c) - (t_1 / c);
	elseif (c <= 3.8e-63)
		tmp = t_0;
	elseif (c <= 240000000000.0)
		tmp = (b / c) + (-1.0 / (c / t_1));
	elseif (c <= 7.2e+47)
		tmp = t_0;
	else
		tmp = (b / c) - (d / (c * (c / a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * N[(N[(1.0 / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -4e-9], N[(N[(b / c), $MachinePrecision] - N[(t$95$1 / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.8e-63], t$95$0, If[LessEqual[c, 240000000000.0], N[(N[(b / c), $MachinePrecision] + N[(-1.0 / N[(c / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 7.2e+47], t$95$0, N[(N[(b / c), $MachinePrecision] - N[(d / N[(c * N[(c / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -4.00000000000000025e-9

   1. Initial program 54.6%

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

   2. Taylor expanded in c around inf 80.4%

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

      1. +-commutative 80.4%

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

      2. mul-1-neg 80.4%

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

      3. unsub-neg 80.4%

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

      4. *-commutative 80.4%

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

      5. associate-/l* 80.8%

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

   4. Simplified 80.8%

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

      1. *-un-lft-identity 80.8%

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

      2. add-sqr-sqrt 42.9%

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

      3. times-frac 42.9%

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

      4. sqrt-div 42.9%

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

      5. unpow2 42.9%

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

      6. sqrt-prod 0.0%

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

      7. add-sqr-sqrt 39.5%

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

      8. sqrt-div 39.5%

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

      9. unpow2 39.5%

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

      10. sqrt-prod 0.0%

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

      11. add-sqr-sqrt 49.2%

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

   6. Applied egg-rr 49.2%

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

      1. associate-*l/ 49.2%

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

      2. *-lft-identity 49.2%

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

      3. associate-/r/ 49.2%

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

   8. Simplified 49.2%

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

      1. sub-neg 49.2%

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

      2. associate-/r/ 46.2%

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

   10. Applied egg-rr 46.2%

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

       1. sub-neg 46.2%

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

       2. associate-*l/ 49.2%

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

       3. associate-*l* 49.2%

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

       4. rem-square-sqrt 86.1%

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

   12. Simplified 86.1%

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

   if -4.00000000000000025e-9 < c < 3.80000000000000017e-63 or 2.4e11 < c < 7.20000000000000015e47

   1. Initial program 72.4%

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

   2. Taylor expanded in c around 0 76.7%

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

      1. +-commutative 76.7%

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

      2. mul-1-neg 76.7%

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

      3. unsub-neg 76.7%

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

      4. associate-/l* 79.4%

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

      5. associate-/r/ 73.8%

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

   4. Simplified 73.8%

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

      1. *-un-lft-identity 73.8%

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

      2. pow2 73.8%

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

      3. times-frac 78.8%

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

   6. Applied egg-rr 78.8%

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

   if 3.80000000000000017e-63 < c < 2.4e11

   1. Initial program 94.1%

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

   2. Taylor expanded in c around inf 69.9%

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

      1. +-commutative 69.9%

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

      2. mul-1-neg 69.9%

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

      3. unsub-neg 69.9%

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

      4. *-commutative 69.9%

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

      5. associate-/l* 64.0%

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

   4. Simplified 64.0%

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

      1. *-un-lft-identity 64.0%

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

      2. add-sqr-sqrt 34.2%

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

      3. times-frac 34.2%

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

      4. sqrt-div 34.4%

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

      5. unpow2 34.4%

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

      6. sqrt-prod 34.4%

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

      7. add-sqr-sqrt 34.4%

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

      8. sqrt-div 40.1%

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

      9. unpow2 40.1%

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

      10. sqrt-prod 40.2%

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

      11. add-sqr-sqrt 40.1%

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

   6. Applied egg-rr 40.1%

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

      1. associate-*l/ 40.1%

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

      2. *-lft-identity 40.1%

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

      3. associate-/r/ 40.1%

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

   8. Simplified 40.1%

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

      1. clear-num 40.3%

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

      2. inv-pow 40.3%

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

   10. Applied egg-rr 40.3%

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

       1. unpow-1 40.3%

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

       2. associate-/l/ 40.4%

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

       3. associate-*l* 40.3%

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

       4. rem-square-sqrt 70.1%

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

   12. Simplified 70.1%

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

   if 7.20000000000000015e47 < c

   1. Initial program 53.0%

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

   2. Taylor expanded in c around inf 91.3%

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

      1. +-commutative 91.3%

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

      2. mul-1-neg 91.3%

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

      3. unsub-neg 91.3%

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

      4. *-commutative 91.3%

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

      5. associate-/l* 91.5%

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

   4. Simplified 91.5%

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

      1. unpow2 91.5%

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

      2. *-un-lft-identity 91.5%

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

      3. times-frac 93.0%

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

   6. Applied egg-rr 93.0%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4 \cdot 10^{-9}:\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{-63}:\\ \;\;\;\;c \cdot \left(\frac{1}{d} \cdot \frac{b}{d}\right) - \frac{a}{d}\\ \mathbf{elif}\;c \leq 240000000000:\\ \;\;\;\;\frac{b}{c} + \frac{-1}{\frac{c}{a \cdot \frac{d}{c}}}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+47}:\\ \;\;\;\;c \cdot \left(\frac{1}{d} \cdot \frac{b}{d}\right) - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c \cdot \frac{c}{a}}\\ \end{array} \]

Alternative 11: 68.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.66 \cdot 10^{-105} \lor \neg \left(c \leq 2.25 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;-\frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.66e-105) (not (<= c 2.25e-63)))
   (- (/ b c) (/ (* a (/ d c)) c))
   (- (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.66e-105) || !(c <= 2.25e-63)) {
		tmp = (b / c) - ((a * (d / 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 <= (-1.66d-105)) .or. (.not. (c <= 2.25d-63))) then
        tmp = (b / c) - ((a * (d / 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 <= -1.66e-105) || !(c <= 2.25e-63)) {
		tmp = (b / c) - ((a * (d / c)) / c);
	} else {
		tmp = -(a / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.66e-105) or not (c <= 2.25e-63):
		tmp = (b / c) - ((a * (d / c)) / c)
	else:
		tmp = -(a / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.66e-105) || !(c <= 2.25e-63))
		tmp = Float64(Float64(b / c) - Float64(Float64(a * Float64(d / 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 <= -1.66e-105) || ~((c <= 2.25e-63)))
		tmp = (b / c) - ((a * (d / c)) / c);
	else
		tmp = -(a / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.66e-105], N[Not[LessEqual[c, 2.25e-63]], $MachinePrecision]], N[(N[(b / c), $MachinePrecision] - N[(N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], (-N[(a / d), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if c < -1.66000000000000009e-105 or 2.25e-63 < c

   1. Initial program 61.4%

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

   2. Taylor expanded in c around inf 76.8%

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

      1. +-commutative 76.8%

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

      2. mul-1-neg 76.8%

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

      3. unsub-neg 76.8%

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

      4. *-commutative 76.8%

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

      5. associate-/l* 74.7%

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

   4. Simplified 74.7%

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

      1. *-un-lft-identity 74.7%

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

      2. add-sqr-sqrt 40.7%

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

      3. times-frac 40.8%

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

      4. sqrt-div 40.8%

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

      5. unpow2 40.8%

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

      6. sqrt-prod 23.6%

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

      7. add-sqr-sqrt 39.4%

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

      8. sqrt-div 40.1%

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

      9. unpow2 40.1%

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

      10. sqrt-prod 25.3%

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

      11. add-sqr-sqrt 45.8%

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

   6. Applied egg-rr 45.8%

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

      1. associate-*l/ 45.8%

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

      2. *-lft-identity 45.8%

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

      3. associate-/r/ 45.4%

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

   8. Simplified 45.4%

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

      1. sub-neg 45.4%

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

      2. associate-/r/ 44.4%

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

   10. Applied egg-rr 44.4%

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

       1. sub-neg 44.4%

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

       2. associate-*l/ 45.5%

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

       3. associate-*l* 45.5%

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

       4. rem-square-sqrt 78.4%

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

   12. Simplified 78.4%

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

   if -1.66000000000000009e-105 < c < 2.25e-63

   1. Initial program 72.3%

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

   2. Taylor expanded in c around 0 76.3%

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

      1. associate-*r/ 76.3%

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

      2. neg-mul-1 76.3%

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

   4. Simplified 76.3%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.66 \cdot 10^{-105} \lor \neg \left(c \leq 2.25 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;-\frac{a}{d}\\ \end{array} \]

Alternative 12: 62.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.75 \cdot 10^{-105} \lor \neg \left(c \leq 5.2 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;-\frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.75e-105) (not (<= c 5.2e-24))) (/ b c) (- (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.75e-105) || !(c <= 5.2e-24)) {
		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 <= (-1.75d-105)) .or. (.not. (c <= 5.2d-24))) 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 <= -1.75e-105) || !(c <= 5.2e-24)) {
		tmp = b / c;
	} else {
		tmp = -(a / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.75e-105) or not (c <= 5.2e-24):
		tmp = b / c
	else:
		tmp = -(a / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.75e-105) || !(c <= 5.2e-24))
		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 <= -1.75e-105) || ~((c <= 5.2e-24)))
		tmp = b / c;
	else
		tmp = -(a / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.75e-105], N[Not[LessEqual[c, 5.2e-24]], $MachinePrecision]], N[(b / c), $MachinePrecision], (-N[(a / d), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if c < -1.75e-105 or 5.2e-24 < c

   1. Initial program 60.2%

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

   2. Taylor expanded in c around inf 64.5%

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

   if -1.75e-105 < c < 5.2e-24

   1. Initial program 73.8%

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

   2. Taylor expanded in c around 0 73.6%

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

      1. associate-*r/ 73.6%

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

      2. neg-mul-1 73.6%

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

   4. Simplified 73.6%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.75 \cdot 10^{-105} \lor \neg \left(c \leq 5.2 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;-\frac{a}{d}\\ \end{array} \]

Alternative 13: 42.6% 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 65.2%

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

2. Taylor expanded in c around inf 44.3%

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

3. Final simplification 44.3%

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

Developer target: 99.3% accurate, 0.1× speedup

Math

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

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 2023334 
(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))))