Complex division, imag part

Percentage Accurate: 62.0% → 85.5%

Time: 12.3s

Alternatives: 11

Speedup: 1.1×

• could not determine a ground truth

  1. a = -2.9430096583721224e+191
  2. b = -2.8126386348900804e+58
  3. c = 1.1157418975697546e-199
  4. d = 4.543427109132206e-307

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+271}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(b, c, a \cdot \left(-d\right)\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) 2e+271)
   (* (/ 1.0 (hypot c d)) (/ (fma b c (* a (- d))) (hypot c d)))
   (- (/ b c) (* (/ d c) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((b * c) - (a * d)) / ((c * c) + (d * d))) <= 2e+271) {
		tmp = (1.0 / hypot(c, d)) * (fma(b, c, (a * -d)) / hypot(c, d));
	} else {
		tmp = (b / c) - ((d / c) * (a / c));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d))) <= 2e+271)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(b, c, Float64(a * Float64(-d))) / hypot(c, d)));
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(d / c) * Float64(a / c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+271], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(b * c + N[(a * (-d)), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / c), $MachinePrecision] - N[(N[(d / c), $MachinePrecision] * N[(a / c), $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))) < 1.99999999999999991e271

   1. Initial program 80.7%

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

      1. *-un-lft-identity 80.7%

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

      2. add-sqr-sqrt 80.7%

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

      3. times-frac 80.7%

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

      4. hypot-define 80.7%

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

      5. fma-neg 80.7%

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

      6. distribute-rgt-neg-in 80.7%

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

      7. hypot-define 96.7%

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

   4. Applied egg-rr 96.7%

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

   if 1.99999999999999991e271 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 14.8%

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

   3. Taylor expanded in c around inf 52.8%

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

      1. *-commutative 52.8%

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

      2. pow2 52.8%

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

      3. times-frac 67.3%

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

   5. Applied egg-rr 67.3%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+271}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(b, c, a \cdot \left(-d\right)\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{a}{-d}\right)\\ \mathbf{if}\;d \leq -7.6 \cdot 10^{-22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-193}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-125}:\\ \;\;\;\;\frac{b}{c} + \frac{a \cdot d}{c} \cdot \frac{-1}{c}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(b, c, a \cdot \left(-d\right)\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma (/ c (hypot c d)) (/ b (hypot c d)) (/ a (- d)))))
   (if (<= d -7.6e-22)
     t_0
     (if (<= d -8e-193)
       (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))
       (if (<= d 7e-125)
         (+ (/ b c) (* (/ (* a d) c) (/ -1.0 c)))
         (if (<= d 1.2e+48)
           (* (fma b c (* a (- d))) (/ 1.0 (pow (hypot c d) 2.0)))
           t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((c / hypot(c, d)), (b / hypot(c, d)), (a / -d));
	double tmp;
	if (d <= -7.6e-22) {
		tmp = t_0;
	} else if (d <= -8e-193) {
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
	} else if (d <= 7e-125) {
		tmp = (b / c) + (((a * d) / c) * (-1.0 / c));
	} else if (d <= 1.2e+48) {
		tmp = fma(b, c, (a * -d)) * (1.0 / pow(hypot(c, d), 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(a / Float64(-d)))
	tmp = 0.0
	if (d <= -7.6e-22)
		tmp = t_0;
	elseif (d <= -8e-193)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 7e-125)
		tmp = Float64(Float64(b / c) + Float64(Float64(Float64(a * d) / c) * Float64(-1.0 / c)));
	elseif (d <= 1.2e+48)
		tmp = Float64(fma(b, c, Float64(a * Float64(-d))) * Float64(1.0 / (hypot(c, d) ^ 2.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = 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]}, If[LessEqual[d, -7.6e-22], t$95$0, If[LessEqual[d, -8e-193], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7e-125], N[(N[(b / c), $MachinePrecision] + N[(N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.2e+48], N[(N[(b * c + N[(a * (-d)), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7.60000000000000046e-22 or 1.2000000000000001e48 < d

   1. Initial program 48.8%

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

      1. div-sub 48.8%

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

      2. *-commutative 48.8%

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

      3. add-sqr-sqrt 48.8%

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

      4. times-frac 51.7%

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

      5. fma-neg 51.7%

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

      6. hypot-define 51.7%

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

      7. hypot-define 66.2%

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

      8. associate-/l* 72.5%

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

      9. add-sqr-sqrt 72.5%

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

      10. pow2 72.5%

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

      11. hypot-define 72.5%

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

   4. Applied egg-rr 72.5%

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

   5. Taylor expanded in d around inf 89.7%

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

   if -7.60000000000000046e-22 < d < -8.0000000000000004e-193

   1. Initial program 88.6%

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

   if -8.0000000000000004e-193 < d < 6.99999999999999995e-125

   1. Initial program 71.5%

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

   3. Taylor expanded in c around inf 87.9%

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

      1. *-un-lft-identity 87.9%

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

      2. pow2 87.9%

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

      3. times-frac 92.7%

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

      4. *-commutative 92.7%

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

   5. Applied egg-rr 92.7%

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

   if 6.99999999999999995e-125 < d < 1.2000000000000001e48

   1. Initial program 85.6%

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

      1. div-inv 85.7%

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

      2. fma-neg 85.7%

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

      3. distribute-rgt-neg-in 85.7%

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

      4. add-sqr-sqrt 85.7%

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

      5. pow2 85.7%

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

      6. hypot-define 85.7%

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

   4. Applied egg-rr 85.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, c, a \cdot \left(-d\right)\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.6 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{a}{-d}\right)\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-193}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-125}:\\ \;\;\;\;\frac{b}{c} + \frac{a \cdot d}{c} \cdot \frac{-1}{c}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(b, c, a \cdot \left(-d\right)\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{a}{-d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{if}\;c \leq -7.3 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -1.08 \cdot 10^{-190}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 3.3 \cdot 10^{-105}:\\ \;\;\;\;b \cdot \left(\frac{1}{d} \cdot \frac{c}{d}\right) - \frac{a}{d}\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{+140}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))
        (t_1 (- (/ b c) (* d (/ (/ a c) c)))))
   (if (<= c -7.3e+53)
     t_1
     (if (<= c -1.08e-190)
       t_0
       (if (<= c 3.3e-105)
         (- (* b (* (/ 1.0 d) (/ c d))) (/ a d))
         (if (<= c 5.8e+140) 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) - (d * ((a / c) / c));
	double tmp;
	if (c <= -7.3e+53) {
		tmp = t_1;
	} else if (c <= -1.08e-190) {
		tmp = t_0;
	} else if (c <= 3.3e-105) {
		tmp = (b * ((1.0 / d) * (c / d))) - (a / d);
	} else if (c <= 5.8e+140) {
		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) - (d * ((a / c) / c))
    if (c <= (-7.3d+53)) then
        tmp = t_1
    else if (c <= (-1.08d-190)) then
        tmp = t_0
    else if (c <= 3.3d-105) then
        tmp = (b * ((1.0d0 / d) * (c / d))) - (a / d)
    else if (c <= 5.8d+140) 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) - (d * ((a / c) / c));
	double tmp;
	if (c <= -7.3e+53) {
		tmp = t_1;
	} else if (c <= -1.08e-190) {
		tmp = t_0;
	} else if (c <= 3.3e-105) {
		tmp = (b * ((1.0 / d) * (c / d))) - (a / d);
	} else if (c <= 5.8e+140) {
		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) - (d * ((a / c) / c))
	tmp = 0
	if c <= -7.3e+53:
		tmp = t_1
	elif c <= -1.08e-190:
		tmp = t_0
	elif c <= 3.3e-105:
		tmp = (b * ((1.0 / d) * (c / d))) - (a / d)
	elif c <= 5.8e+140:
		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(d * Float64(Float64(a / c) / c)))
	tmp = 0.0
	if (c <= -7.3e+53)
		tmp = t_1;
	elseif (c <= -1.08e-190)
		tmp = t_0;
	elseif (c <= 3.3e-105)
		tmp = Float64(Float64(b * Float64(Float64(1.0 / d) * Float64(c / d))) - Float64(a / d));
	elseif (c <= 5.8e+140)
		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) - (d * ((a / c) / c));
	tmp = 0.0;
	if (c <= -7.3e+53)
		tmp = t_1;
	elseif (c <= -1.08e-190)
		tmp = t_0;
	elseif (c <= 3.3e-105)
		tmp = (b * ((1.0 / d) * (c / d))) - (a / d);
	elseif (c <= 5.8e+140)
		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[(d * N[(N[(a / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -7.3e+53], t$95$1, If[LessEqual[c, -1.08e-190], t$95$0, If[LessEqual[c, 3.3e-105], N[(N[(b * N[(N[(1.0 / d), $MachinePrecision] * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 5.8e+140], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.30000000000000016e53 or 5.7999999999999998e140 < c

   1. Initial program 39.3%

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

   3. Taylor expanded in c around inf 78.6%

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

      1. *-un-lft-identity 78.6%

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

      2. pow2 78.6%

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

      3. times-frac 78.8%

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

      4. *-commutative 78.8%

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

   5. Applied egg-rr 78.8%

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

      1. associate-*l/ 78.8%

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

      2. *-un-lft-identity 78.8%

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

      3. associate-/l* 85.6%

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

   7. Applied egg-rr 85.6%

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

      1. associate-/l* 87.4%

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

   9. Applied egg-rr 87.4%

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

   if -7.30000000000000016e53 < c < -1.08e-190 or 3.2999999999999999e-105 < c < 5.7999999999999998e140

   1. Initial program 84.5%

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

   if -1.08e-190 < c < 3.2999999999999999e-105

   1. Initial program 74.4%

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

      1. *-un-lft-identity 74.4%

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

      2. add-sqr-sqrt 74.4%

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

      3. times-frac 74.4%

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

      4. hypot-define 74.4%

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

      5. fma-neg 74.4%

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

      6. distribute-rgt-neg-in 74.4%

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

      7. hypot-define 87.2%

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

   4. Applied egg-rr 87.2%

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

   5. Taylor expanded in c around 0 88.4%

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

      1. +-commutative 88.4%

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

      2. mul-1-neg 88.4%

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

      3. unsub-neg 88.4%

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

      4. associate-/l* 91.1%

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

   7. Simplified 91.1%

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

      1. *-un-lft-identity 91.1%

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

      2. unpow2 91.1%

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

      3. times-frac 94.8%

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

   9. Applied egg-rr 94.8%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.3 \cdot 10^{+53}:\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -1.08 \cdot 10^{-190}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 3.3 \cdot 10^{-105}:\\ \;\;\;\;b \cdot \left(\frac{1}{d} \cdot \frac{c}{d}\right) - \frac{a}{d}\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{+140}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{-d}\\ \mathbf{if}\;c \leq -2.8 \cdot 10^{-82}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{-80}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2 \cdot 10^{+62}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ a (- d))))
   (if (<= c -2.8e-82)
     (/ b c)
     (if (<= c 2.6e-80)
       t_0
       (if (<= c 5.8e+51)
         (/ (* b c) (+ (* c c) (* d d)))
         (if (<= c 2e+62) t_0 (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (c <= -2.8e-82) {
		tmp = b / c;
	} else if (c <= 2.6e-80) {
		tmp = t_0;
	} else if (c <= 5.8e+51) {
		tmp = (b * c) / ((c * c) + (d * d));
	} else if (c <= 2e+62) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / -d
    if (c <= (-2.8d-82)) then
        tmp = b / c
    else if (c <= 2.6d-80) then
        tmp = t_0
    else if (c <= 5.8d+51) then
        tmp = (b * c) / ((c * c) + (d * d))
    else if (c <= 2d+62) then
        tmp = t_0
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (c <= -2.8e-82) {
		tmp = b / c;
	} else if (c <= 2.6e-80) {
		tmp = t_0;
	} else if (c <= 5.8e+51) {
		tmp = (b * c) / ((c * c) + (d * d));
	} else if (c <= 2e+62) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = a / -d
	tmp = 0
	if c <= -2.8e-82:
		tmp = b / c
	elif c <= 2.6e-80:
		tmp = t_0
	elif c <= 5.8e+51:
		tmp = (b * c) / ((c * c) + (d * d))
	elif c <= 2e+62:
		tmp = t_0
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(a / Float64(-d))
	tmp = 0.0
	if (c <= -2.8e-82)
		tmp = Float64(b / c);
	elseif (c <= 2.6e-80)
		tmp = t_0;
	elseif (c <= 5.8e+51)
		tmp = Float64(Float64(b * c) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 2e+62)
		tmp = t_0;
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = a / -d;
	tmp = 0.0;
	if (c <= -2.8e-82)
		tmp = b / c;
	elseif (c <= 2.6e-80)
		tmp = t_0;
	elseif (c <= 5.8e+51)
		tmp = (b * c) / ((c * c) + (d * d));
	elseif (c <= 2e+62)
		tmp = t_0;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(a / (-d)), $MachinePrecision]}, If[LessEqual[c, -2.8e-82], N[(b / c), $MachinePrecision], If[LessEqual[c, 2.6e-80], t$95$0, If[LessEqual[c, 5.8e+51], N[(N[(b * c), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2e+62], t$95$0, N[(b / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.80000000000000024e-82 or 2.00000000000000007e62 < c

   1. Initial program 51.5%

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

   3. Taylor expanded in c around inf 67.5%

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

   if -2.80000000000000024e-82 < c < 2.6000000000000001e-80 or 5.7999999999999997e51 < c < 2.00000000000000007e62

   1. Initial program 77.7%

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

   3. Taylor expanded in c around 0 73.6%

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

      1. associate-*r/ 73.6%

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

      2. neg-mul-1 73.6%

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

   5. Simplified 73.6%

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

   if 2.6000000000000001e-80 < c < 5.7999999999999997e51

   1. Initial program 95.3%

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

   3. Taylor expanded in b around inf 57.7%

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

      1. *-commutative 57.7%

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

   5. Simplified 57.7%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.8 \cdot 10^{-82}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{-80}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -0.0034 \lor \neg \left(c \leq 2.5 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\frac{1}{d} \cdot \frac{c}{d}\right) - \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -0.0034) (not (<= c 2.5e-44)))
   (- (/ b c) (* d (/ (/ a c) c)))
   (- (* b (* (/ 1.0 d) (/ c d))) (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -0.0034) || !(c <= 2.5e-44)) {
		tmp = (b / c) - (d * ((a / c) / c));
	} else {
		tmp = (b * ((1.0 / d) * (c / 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 <= (-0.0034d0)) .or. (.not. (c <= 2.5d-44))) then
        tmp = (b / c) - (d * ((a / c) / c))
    else
        tmp = (b * ((1.0d0 / d) * (c / 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 <= -0.0034) || !(c <= 2.5e-44)) {
		tmp = (b / c) - (d * ((a / c) / c));
	} else {
		tmp = (b * ((1.0 / d) * (c / d))) - (a / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -0.0034) or not (c <= 2.5e-44):
		tmp = (b / c) - (d * ((a / c) / c))
	else:
		tmp = (b * ((1.0 / d) * (c / d))) - (a / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -0.0034) || !(c <= 2.5e-44))
		tmp = Float64(Float64(b / c) - Float64(d * Float64(Float64(a / c) / c)));
	else
		tmp = Float64(Float64(b * Float64(Float64(1.0 / d) * Float64(c / d))) - Float64(a / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -0.0034) || ~((c <= 2.5e-44)))
		tmp = (b / c) - (d * ((a / c) / c));
	else
		tmp = (b * ((1.0 / d) * (c / d))) - (a / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -0.0034], N[Not[LessEqual[c, 2.5e-44]], $MachinePrecision]], N[(N[(b / c), $MachinePrecision] - N[(d * N[(N[(a / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(N[(1.0 / d), $MachinePrecision] * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -0.00339999999999999981 or 2.50000000000000019e-44 < c

   1. Initial program 53.0%

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

   3. Taylor expanded in c around inf 73.5%

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

      1. *-un-lft-identity 73.5%

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

      2. pow2 73.5%

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

      3. times-frac 73.6%

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

      4. *-commutative 73.6%

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

   5. Applied egg-rr 73.6%

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

      1. associate-*l/ 73.6%

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

      2. *-un-lft-identity 73.6%

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

      3. associate-/l* 78.1%

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

   7. Applied egg-rr 78.1%

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

      1. associate-/l* 79.2%

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

   9. Applied egg-rr 79.2%

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

   if -0.00339999999999999981 < c < 2.50000000000000019e-44

   1. Initial program 80.2%

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

      1. *-un-lft-identity 80.2%

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

      2. add-sqr-sqrt 80.2%

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

      3. times-frac 80.3%

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

      4. hypot-define 80.3%

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

      5. fma-neg 80.3%

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

      6. distribute-rgt-neg-in 80.3%

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

      7. hypot-define 90.4%

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

   4. Applied egg-rr 90.4%

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

   5. Taylor expanded in c around 0 82.1%

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

      1. +-commutative 82.1%

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

      2. mul-1-neg 82.1%

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

      3. unsub-neg 82.1%

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

      4. associate-/l* 83.0%

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

   7. Simplified 83.0%

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

      1. *-un-lft-identity 83.0%

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

      2. unpow2 83.0%

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

      3. times-frac 85.3%

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

   9. Applied egg-rr 85.3%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -0.0034 \lor \neg \left(c \leq 2.5 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\frac{1}{d} \cdot \frac{c}{d}\right) - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{b}{c} + \frac{a \cdot d}{c} \cdot \frac{-1}{c}\\ \mathbf{elif}\;c \leq 6.6 \cdot 10^{-44}:\\ \;\;\;\;b \cdot \left(\frac{1}{d} \cdot \frac{c}{d}\right) - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -7.5e-74)
   (+ (/ b c) (* (/ (* a d) c) (/ -1.0 c)))
   (if (<= c 6.6e-44)
     (- (* b (* (/ 1.0 d) (/ c d))) (/ a d))
     (- (/ b c) (* d (/ (/ a c) c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7.5e-74) {
		tmp = (b / c) + (((a * d) / c) * (-1.0 / c));
	} else if (c <= 6.6e-44) {
		tmp = (b * ((1.0 / d) * (c / d))) - (a / d);
	} else {
		tmp = (b / c) - (d * ((a / c) / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-7.5d-74)) then
        tmp = (b / c) + (((a * d) / c) * ((-1.0d0) / c))
    else if (c <= 6.6d-44) then
        tmp = (b * ((1.0d0 / d) * (c / d))) - (a / d)
    else
        tmp = (b / c) - (d * ((a / c) / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7.5e-74) {
		tmp = (b / c) + (((a * d) / c) * (-1.0 / c));
	} else if (c <= 6.6e-44) {
		tmp = (b * ((1.0 / d) * (c / d))) - (a / d);
	} else {
		tmp = (b / c) - (d * ((a / c) / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -7.5e-74:
		tmp = (b / c) + (((a * d) / c) * (-1.0 / c))
	elif c <= 6.6e-44:
		tmp = (b * ((1.0 / d) * (c / d))) - (a / d)
	else:
		tmp = (b / c) - (d * ((a / c) / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -7.5e-74)
		tmp = Float64(Float64(b / c) + Float64(Float64(Float64(a * d) / c) * Float64(-1.0 / c)));
	elseif (c <= 6.6e-44)
		tmp = Float64(Float64(b * Float64(Float64(1.0 / d) * Float64(c / d))) - Float64(a / d));
	else
		tmp = Float64(Float64(b / c) - Float64(d * Float64(Float64(a / c) / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -7.5e-74)
		tmp = (b / c) + (((a * d) / c) * (-1.0 / c));
	elseif (c <= 6.6e-44)
		tmp = (b * ((1.0 / d) * (c / d))) - (a / d);
	else
		tmp = (b / c) - (d * ((a / c) / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -7.5e-74], N[(N[(b / c), $MachinePrecision] + N[(N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6.6e-44], N[(N[(b * N[(N[(1.0 / d), $MachinePrecision] * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], N[(N[(b / c), $MachinePrecision] - N[(d * N[(N[(a / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.5e-74

   1. Initial program 58.3%

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

   3. Taylor expanded in c around inf 73.6%

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

      1. *-un-lft-identity 73.6%

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

      2. pow2 73.6%

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

      3. times-frac 74.8%

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

      4. *-commutative 74.8%

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

   5. Applied egg-rr 74.8%

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

   if -7.5e-74 < c < 6.60000000000000011e-44

   1. Initial program 80.4%

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

      1. *-un-lft-identity 80.4%

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

      2. add-sqr-sqrt 80.4%

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

      3. times-frac 80.4%

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

      4. hypot-define 80.4%

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

      5. fma-neg 80.4%

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

      6. distribute-rgt-neg-in 80.4%

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

      7. hypot-define 90.2%

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

   4. Applied egg-rr 90.2%

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

   5. Taylor expanded in c around 0 87.7%

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

      1. +-commutative 87.7%

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

      2. mul-1-neg 87.7%

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

      3. unsub-neg 87.7%

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

      4. associate-/l* 88.6%

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

   7. Simplified 88.6%

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

      1. *-un-lft-identity 88.6%

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

      2. unpow2 88.6%

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

      3. times-frac 91.2%

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

   9. Applied egg-rr 91.2%

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

   if 6.60000000000000011e-44 < c

   1. Initial program 51.9%

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

   3. Taylor expanded in c around inf 70.2%

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

      1. *-un-lft-identity 70.2%

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

      2. pow2 70.2%

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

      3. times-frac 68.9%

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

      4. *-commutative 68.9%

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

   5. Applied egg-rr 68.9%

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

      1. associate-*l/ 68.9%

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

      2. *-un-lft-identity 68.9%

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

      3. associate-/l* 76.4%

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

   7. Applied egg-rr 76.4%

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

      1. associate-/l* 77.1%

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

   9. Applied egg-rr 77.1%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{b}{c} + \frac{a \cdot d}{c} \cdot \frac{-1}{c}\\ \mathbf{elif}\;c \leq 6.6 \cdot 10^{-44}:\\ \;\;\;\;b \cdot \left(\frac{1}{d} \cdot \frac{c}{d}\right) - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.05 \cdot 10^{-105} \lor \neg \left(c \leq 9 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -3.05e-105) (not (<= c 9e-44)))
   (- (/ b c) (* d (/ (/ a c) c)))
   (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.05e-105) || !(c <= 9e-44)) {
		tmp = (b / c) - (d * ((a / c) / c));
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-3.05d-105)) .or. (.not. (c <= 9d-44))) then
        tmp = (b / c) - (d * ((a / c) / c))
    else
        tmp = a / -d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.05e-105) || !(c <= 9e-44)) {
		tmp = (b / c) - (d * ((a / c) / c));
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -3.05e-105) or not (c <= 9e-44):
		tmp = (b / c) - (d * ((a / c) / c))
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -3.05e-105) || !(c <= 9e-44))
		tmp = Float64(Float64(b / c) - Float64(d * Float64(Float64(a / c) / c)));
	else
		tmp = Float64(a / Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -3.05e-105) || ~((c <= 9e-44)))
		tmp = (b / c) - (d * ((a / c) / c));
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -3.05e-105], N[Not[LessEqual[c, 9e-44]], $MachinePrecision]], N[(N[(b / c), $MachinePrecision] - N[(d * N[(N[(a / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.04999999999999992e-105 or 8.9999999999999997e-44 < c

   1. Initial program 57.5%

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

   3. Taylor expanded in c around inf 70.9%

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

      1. *-un-lft-identity 70.9%

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

      2. pow2 70.9%

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

      3. times-frac 70.9%

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

      4. *-commutative 70.9%

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

   5. Applied egg-rr 70.9%

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

      1. associate-*l/ 71.0%

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

      2. *-un-lft-identity 71.0%

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

      3. associate-/l* 72.8%

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

   7. Applied egg-rr 72.8%

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

      1. associate-/l* 73.8%

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

   9. Applied egg-rr 73.8%

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

   if -3.04999999999999992e-105 < c < 8.9999999999999997e-44

   1. Initial program 79.1%

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

   3. Taylor expanded in c around 0 72.2%

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

      1. associate-*r/ 72.2%

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

      2. neg-mul-1 72.2%

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

   5. Simplified 72.2%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.05 \cdot 10^{-105} \lor \neg \left(c \leq 9 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -7.2e-103)
   (- (/ b c) (* d (/ (/ a c) c)))
   (if (<= c 9.5e-61) (/ a (- d)) (- (/ b c) (* (/ d c) (/ a c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7.2e-103) {
		tmp = (b / c) - (d * ((a / c) / c));
	} else if (c <= 9.5e-61) {
		tmp = a / -d;
	} else {
		tmp = (b / c) - ((d / c) * (a / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-7.2d-103)) then
        tmp = (b / c) - (d * ((a / c) / c))
    else if (c <= 9.5d-61) then
        tmp = a / -d
    else
        tmp = (b / c) - ((d / c) * (a / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7.2e-103) {
		tmp = (b / c) - (d * ((a / c) / c));
	} else if (c <= 9.5e-61) {
		tmp = a / -d;
	} else {
		tmp = (b / c) - ((d / c) * (a / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -7.2e-103:
		tmp = (b / c) - (d * ((a / c) / c))
	elif c <= 9.5e-61:
		tmp = a / -d
	else:
		tmp = (b / c) - ((d / c) * (a / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -7.2e-103)
		tmp = Float64(Float64(b / c) - Float64(d * Float64(Float64(a / c) / c)));
	elseif (c <= 9.5e-61)
		tmp = Float64(a / Float64(-d));
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(d / c) * Float64(a / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -7.2e-103)
		tmp = (b / c) - (d * ((a / c) / c));
	elseif (c <= 9.5e-61)
		tmp = a / -d;
	else
		tmp = (b / c) - ((d / c) * (a / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -7.2e-103], N[(N[(b / c), $MachinePrecision] - N[(d * N[(N[(a / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 9.5e-61], N[(a / (-d)), $MachinePrecision], N[(N[(b / c), $MachinePrecision] - N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.1999999999999996e-103

   1. Initial program 61.6%

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

   3. Taylor expanded in c around inf 71.4%

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

      1. *-un-lft-identity 71.4%

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

      2. pow2 71.4%

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

      3. times-frac 72.5%

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

      4. *-commutative 72.5%

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

   5. Applied egg-rr 72.5%

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

      1. associate-*l/ 72.5%

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

      2. *-un-lft-identity 72.5%

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

      3. associate-/l* 70.2%

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

   7. Applied egg-rr 70.2%

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

      1. associate-/l* 71.4%

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

   9. Applied egg-rr 71.4%

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

   if -7.1999999999999996e-103 < c < 9.49999999999999986e-61

   1. Initial program 78.1%

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

   3. Taylor expanded in c around 0 73.6%

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

      1. associate-*r/ 73.6%

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

      2. neg-mul-1 73.6%

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

   5. Simplified 73.6%

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

   if 9.49999999999999986e-61 < c

   1. Initial program 55.3%

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

   3. Taylor expanded in c around inf 69.6%

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

      1. *-commutative 69.6%

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

      2. pow2 69.6%

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

      3. times-frac 75.4%

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

   5. Applied egg-rr 75.4%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.2 \cdot 10^{-82} \lor \neg \left(c \leq 2.9 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.2e-82) (not (<= c 2.9e-9))) (/ b c) (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.2e-82) || !(c <= 2.9e-9)) {
		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 <= (-1.2d-82)) .or. (.not. (c <= 2.9d-9))) 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 <= -1.2e-82) || !(c <= 2.9e-9)) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.2e-82) or not (c <= 2.9e-9):
		tmp = b / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.2e-82) || !(c <= 2.9e-9))
		tmp = Float64(b / c);
	else
		tmp = Float64(a / Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.2e-82) || ~((c <= 2.9e-9)))
		tmp = b / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.2e-82], N[Not[LessEqual[c, 2.9e-9]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.20000000000000004e-82 or 2.89999999999999991e-9 < c

   1. Initial program 54.9%

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

   3. Taylor expanded in c around inf 65.3%

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

   if -1.20000000000000004e-82 < c < 2.89999999999999991e-9

   1. Initial program 80.2%

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

   3. Taylor expanded in c around 0 69.0%

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

      1. associate-*r/ 69.0%

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

      2. neg-mul-1 69.0%

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

   5. Simplified 69.0%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.2 \cdot 10^{-82} \lor \neg \left(c \leq 2.9 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.3 \cdot 10^{+166} \lor \neg \left(d \leq 1.36 \cdot 10^{+167}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -3.3e+166) (not (<= d 1.36e+167))) (/ a d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3.3e+166) || !(d <= 1.36e+167)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-3.3d+166)) .or. (.not. (d <= 1.36d+167))) then
        tmp = a / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3.3e+166) || !(d <= 1.36e+167)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -3.3e+166) or not (d <= 1.36e+167):
		tmp = a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -3.3e+166) || !(d <= 1.36e+167))
		tmp = Float64(a / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -3.3e+166) || ~((d <= 1.36e+167)))
		tmp = a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -3.3e+166], N[Not[LessEqual[d, 1.36e+167]], $MachinePrecision]], N[(a / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -3.3000000000000002e166 or 1.36e167 < d

   1. Initial program 36.9%

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

   3. Taylor expanded in b around 0 37.2%

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

      1. mul-1-neg 37.2%

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

      2. distribute-rgt-neg-out 37.2%

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

   5. Simplified 37.2%

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

      1. *-commutative 37.2%

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

      2. add-sqr-sqrt 37.2%

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

      3. hypot-undefine 37.2%

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

      4. hypot-undefine 37.2%

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

      5. times-frac 83.0%

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

      6. add-sqr-sqrt 41.2%

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

      7. sqrt-unprod 0.9%

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

      8. sqr-neg 0.9%

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

      9. sqrt-prod 15.9%

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

      10. add-sqr-sqrt 37.3%

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

      11. hypot-undefine 38.1%

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

      12. +-commutative 38.1%

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

      13. hypot-define 37.3%

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

      14. hypot-undefine 38.1%

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

      15. +-commutative 38.1%

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

      16. hypot-define 37.3%

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

   7. Applied egg-rr 37.3%

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

   8. Taylor expanded in d around inf 35.9%

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

   if -3.3000000000000002e166 < d < 1.36e167

   1. Initial program 75.9%

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

   3. Taylor expanded in c around inf 52.7%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.3 \cdot 10^{+166} \lor \neg \left(d \leq 1.36 \cdot 10^{+167}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 10.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{a}{d} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.3%

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

3. Taylor expanded in b around 0 43.2%

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

   1. mul-1-neg 43.2%

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

   2. distribute-rgt-neg-out 43.2%

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

5. Simplified 43.2%

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

   1. *-commutative 43.2%

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

   2. add-sqr-sqrt 43.2%

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

   3. hypot-undefine 43.2%

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

   4. hypot-undefine 43.2%

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

   5. times-frac 55.9%

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

   6. add-sqr-sqrt 25.9%

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

   7. sqrt-unprod 17.6%

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

   8. sqr-neg 17.6%

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

   9. sqrt-prod 8.9%

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

   10. add-sqr-sqrt 16.9%

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

   11. hypot-undefine 17.8%

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

   12. +-commutative 17.8%

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

   13. hypot-define 16.9%

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

   14. hypot-undefine 17.8%

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

   15. +-commutative 17.8%

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

   16. hypot-define 16.9%

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

7. Applied egg-rr 16.9%

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

8. Taylor expanded in d around inf 11.4%

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

9. Final simplification 11.4%

   \[\leadsto \frac{a}{d} \]
10. Add Preprocessing

Developer target: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024046 
(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))))