Complex division, imag part

Percentage Accurate: 62.0% → 92.7%

Time: 9.9s

Alternatives: 12

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: 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: 92.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ t_2 := \mathsf{fma}\left(t_0, t_1, \frac{-a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)\\ t_3 := \mathsf{fma}\left(t_0, t_1, -\frac{a}{d}\right)\\ \mathbf{if}\;d \leq -2 \cdot 10^{+137}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -7.6 \cdot 10^{-131}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{-154}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 2.85 \cdot 10^{+167}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ c (hypot c d)))
        (t_1 (/ b (hypot c d)))
        (t_2 (fma t_0 t_1 (/ (- a) (/ (pow (hypot c d) 2.0) d))))
        (t_3 (fma t_0 t_1 (- (/ a d)))))
   (if (<= d -2e+137)
     t_3
     (if (<= d -7.6e-131)
       t_2
       (if (<= d 1.85e-154)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 2.85e+167) t_2 t_3))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = c / hypot(c, d);
	double t_1 = b / hypot(c, d);
	double t_2 = fma(t_0, t_1, (-a / (pow(hypot(c, d), 2.0) / d)));
	double t_3 = fma(t_0, t_1, -(a / d));
	double tmp;
	if (d <= -2e+137) {
		tmp = t_3;
	} else if (d <= -7.6e-131) {
		tmp = t_2;
	} else if (d <= 1.85e-154) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 2.85e+167) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(c / hypot(c, d))
	t_1 = Float64(b / hypot(c, d))
	t_2 = fma(t_0, t_1, Float64(Float64(-a) / Float64((hypot(c, d) ^ 2.0) / d)))
	t_3 = fma(t_0, t_1, Float64(-Float64(a / d)))
	tmp = 0.0
	if (d <= -2e+137)
		tmp = t_3;
	elseif (d <= -7.6e-131)
		tmp = t_2;
	elseif (d <= 1.85e-154)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 2.85e+167)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1 + N[((-a) / N[(N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * t$95$1 + (-N[(a / d), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[d, -2e+137], t$95$3, If[LessEqual[d, -7.6e-131], t$95$2, If[LessEqual[d, 1.85e-154], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.85e+167], t$95$2, t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.0000000000000001e137 or 2.85000000000000008e167 < d

   1. Initial program 34.7%

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

      1. div-sub 34.7%

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

      2. *-commutative 34.7%

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

      3. add-sqr-sqrt 34.7%

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

      4. times-frac 37.9%

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

      5. fma-neg 37.9%

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

      6. hypot-def 37.9%

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

      7. hypot-def 53.2%

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

      8. associate-/l* 56.1%

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

      9. add-sqr-sqrt 56.1%

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

      10. pow2 56.1%

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

      11. hypot-def 56.1%

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

   3. Applied egg-rr 56.1%

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

   4. Taylor expanded in c around 0 91.6%

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

   if -2.0000000000000001e137 < d < -7.59999999999999989e-131 or 1.84999999999999993e-154 < d < 2.85000000000000008e167

   1. Initial program 67.7%

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

      1. div-sub 67.7%

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

      2. *-commutative 67.7%

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

      3. add-sqr-sqrt 67.7%

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

      4. times-frac 72.5%

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

      5. fma-neg 72.5%

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

      6. hypot-def 72.5%

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

      7. hypot-def 88.3%

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

      8. associate-/l* 96.3%

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

      9. add-sqr-sqrt 96.3%

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

      10. pow2 96.3%

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

      11. hypot-def 96.3%

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

   3. Applied egg-rr 96.3%

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

   if -7.59999999999999989e-131 < d < 1.84999999999999993e-154

   1. Initial program 66.2%

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

   2. Taylor expanded in c around inf 76.6%

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

      1. +-commutative 76.6%

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

      2. mul-1-neg 76.6%

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

      3. unsub-neg 76.6%

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

      4. unpow2 76.6%

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

      5. times-frac 84.5%

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

   4. Simplified 84.5%

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

      1. associate-*l/ 84.5%

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

      2. sub-div 88.0%

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

   6. Applied egg-rr 88.0%

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

      1. *-commutative 88.0%

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

      2. associate-*l/ 90.6%

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

   8. Applied egg-rr 90.6%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{a}{d}\right)\\ \mathbf{elif}\;d \leq -7.6 \cdot 10^{-131}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{-154}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 2.85 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{a}{d}\right)\\ \end{array} \]

Alternative 2: 89.2% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c b) (* d a))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 1e+302)
     (* (/ 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 = (c * b) - (d * a);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+302) {
		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(c * b) - Float64(d * a))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= 1e+302)
		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[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+302], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(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))) < 1.0000000000000001e302

   1. Initial program 79.3%

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

      1. *-un-lft-identity 79.3%

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

      2. add-sqr-sqrt 79.3%

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

      3. times-frac 79.3%

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

      4. hypot-def 79.3%

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

      5. hypot-def 95.2%

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

   3. Applied egg-rr 95.2%

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

   if 1.0000000000000001e302 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 6.8%

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

      1. div-sub 5.2%

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

      2. *-commutative 5.2%

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

      3. add-sqr-sqrt 5.2%

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

      4. times-frac 14.4%

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

      5. fma-neg 14.4%

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

      6. hypot-def 14.4%

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

      7. hypot-def 46.1%

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

      8. associate-/l* 60.5%

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

      9. add-sqr-sqrt 60.5%

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

      10. pow2 60.5%

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

      11. hypot-def 60.5%

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

   3. Applied egg-rr 60.5%

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

   4. Taylor expanded in c around 0 67.2%

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

4. Final simplification 87.2%

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

Alternative 3: 85.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot b - d \cdot a\\ t_1 := \frac{t_0}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_1 \leq 10^{+302}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c b) (* d a))) (t_1 (/ t_0 (+ (* c c) (* d d)))))
   (if (<= t_1 1e+302)
     (* (/ 1.0 (hypot c d)) (/ t_0 (hypot c d)))
     (if (<= t_1 INFINITY)
       (- (* (/ c d) (/ b d)) (/ a d))
       (/ (- b (* a (/ d c))) c)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * b) - (d * a);
	double t_1 = t_0 / ((c * c) + (d * d));
	double tmp;
	if (t_1 <= 1e+302) {
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((c / d) * (b / d)) - (a / d);
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (c * b) - (d * a);
	double t_1 = t_0 / ((c * c) + (d * d));
	double tmp;
	if (t_1 <= 1e+302) {
		tmp = (1.0 / Math.hypot(c, d)) * (t_0 / Math.hypot(c, d));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((c / d) * (b / d)) - (a / d);
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (c * b) - (d * a)
	t_1 = t_0 / ((c * c) + (d * d))
	tmp = 0
	if t_1 <= 1e+302:
		tmp = (1.0 / math.hypot(c, d)) * (t_0 / math.hypot(c, d))
	elif t_1 <= math.inf:
		tmp = ((c / d) * (b / d)) - (a / d)
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * b) - Float64(d * a))
	t_1 = Float64(t_0 / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (t_1 <= 1e+302)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, d)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(c / d) * Float64(b / d)) - Float64(a / d));
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (c * b) - (d * a);
	t_1 = t_0 / ((c * c) + (d * d));
	tmp = 0.0;
	if (t_1 <= 1e+302)
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	elseif (t_1 <= Inf)
		tmp = ((c / d) * (b / d)) - (a / d);
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+302], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 1.0000000000000001e302

   1. Initial program 79.3%

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

      1. *-un-lft-identity 79.3%

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

      2. add-sqr-sqrt 79.3%

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

      3. times-frac 79.3%

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

      4. hypot-def 79.3%

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

      5. hypot-def 95.2%

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

   3. Applied egg-rr 95.2%

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

   if 1.0000000000000001e302 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < +inf.0

   1. Initial program 22.5%

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

   2. Taylor expanded in c around 0 60.8%

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

      1. +-commutative 60.8%

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

      2. mul-1-neg 60.8%

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

      3. unsub-neg 60.8%

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

      4. *-commutative 60.8%

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

      5. unpow2 60.8%

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

      6. times-frac 70.2%

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

   4. Simplified 70.2%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in c around inf 45.0%

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

      1. +-commutative 45.0%

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

      2. mul-1-neg 45.0%

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

      3. unsub-neg 45.0%

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

      4. unpow2 45.0%

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

      5. times-frac 59.4%

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

   4. Simplified 59.4%

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

      1. associate-*l/ 59.4%

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

      2. sub-div 59.5%

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

   6. Applied egg-rr 59.5%

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

4. Final simplification 85.9%

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

Alternative 4: 79.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{if}\;c \leq -4.4 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 3.3 \cdot 10^{+100}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* d (/ a c))) c)))
   (if (<= c -4.4e+52)
     t_0
     (if (<= c 5.8e-92)
       (- (* (/ c d) (/ b d)) (/ a d))
       (if (<= c 3.3e+100) (/ (- (* c b) (* d a)) (fma c c (* d d))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - (d * (a / c))) / c;
	double tmp;
	if (c <= -4.4e+52) {
		tmp = t_0;
	} else if (c <= 5.8e-92) {
		tmp = ((c / d) * (b / d)) - (a / d);
	} else if (c <= 3.3e+100) {
		tmp = ((c * b) - (d * a)) / fma(c, c, (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(d * Float64(a / c))) / c)
	tmp = 0.0
	if (c <= -4.4e+52)
		tmp = t_0;
	elseif (c <= 5.8e-92)
		tmp = Float64(Float64(Float64(c / d) * Float64(b / d)) - Float64(a / d));
	elseif (c <= 3.3e+100)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / fma(c, c, Float64(d * d)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -4.4e+52], t$95$0, If[LessEqual[c, 5.8e-92], N[(N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.3e+100], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -4.4e52 or 3.3000000000000001e100 < c

   1. Initial program 39.7%

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

   2. Taylor expanded in c around inf 76.0%

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

      1. +-commutative 76.0%

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

      2. mul-1-neg 76.0%

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

      3. unsub-neg 76.0%

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

      4. unpow2 76.0%

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

      5. times-frac 85.5%

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

   4. Simplified 85.5%

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

      1. associate-*r/ 86.7%

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

      2. sub-div 86.7%

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

   6. Applied egg-rr 86.7%

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

   if -4.4e52 < c < 5.79999999999999969e-92

   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 77.3%

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

      1. +-commutative 77.3%

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

      2. mul-1-neg 77.3%

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

      3. unsub-neg 77.3%

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

      4. *-commutative 77.3%

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

      5. unpow2 77.3%

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

      6. times-frac 82.0%

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

   4. Simplified 82.0%

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

   if 5.79999999999999969e-92 < c < 3.3000000000000001e100

   1. Initial program 80.1%

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

      1. fma-def 80.2%

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

   3. Simplified 80.2%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.4 \cdot 10^{+52}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 3.3 \cdot 10^{+100}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]

Alternative 5: 66.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{a}{d}\\ t_1 := \frac{b - a \cdot \frac{d}{c}}{c}\\ t_2 := \frac{c}{d} \cdot \frac{b}{d}\\ \mathbf{if}\;c \leq -2.05 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -1.04 \cdot 10^{-54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -2.2 \cdot 10^{-105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{-113}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 2400:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{+26} \lor \neg \left(c \leq 9.5 \cdot 10^{+99}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (/ a d)))
        (t_1 (/ (- b (* a (/ d c))) c))
        (t_2 (* (/ c d) (/ b d))))
   (if (<= c -2.05e+19)
     t_1
     (if (<= c -1.04e-54)
       t_0
       (if (<= c -2.2e-105)
         t_1
         (if (<= c -1.1e-113)
           t_2
           (if (<= c 2400.0)
             t_0
             (if (or (<= c 5.2e+26) (not (<= c 9.5e+99))) t_1 t_2))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -(a / d);
	double t_1 = (b - (a * (d / c))) / c;
	double t_2 = (c / d) * (b / d);
	double tmp;
	if (c <= -2.05e+19) {
		tmp = t_1;
	} else if (c <= -1.04e-54) {
		tmp = t_0;
	} else if (c <= -2.2e-105) {
		tmp = t_1;
	} else if (c <= -1.1e-113) {
		tmp = t_2;
	} else if (c <= 2400.0) {
		tmp = t_0;
	} else if ((c <= 5.2e+26) || !(c <= 9.5e+99)) {
		tmp = t_1;
	} 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 = -(a / d)
    t_1 = (b - (a * (d / c))) / c
    t_2 = (c / d) * (b / d)
    if (c <= (-2.05d+19)) then
        tmp = t_1
    else if (c <= (-1.04d-54)) then
        tmp = t_0
    else if (c <= (-2.2d-105)) then
        tmp = t_1
    else if (c <= (-1.1d-113)) then
        tmp = t_2
    else if (c <= 2400.0d0) then
        tmp = t_0
    else if ((c <= 5.2d+26) .or. (.not. (c <= 9.5d+99))) then
        tmp = t_1
    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 = -(a / d);
	double t_1 = (b - (a * (d / c))) / c;
	double t_2 = (c / d) * (b / d);
	double tmp;
	if (c <= -2.05e+19) {
		tmp = t_1;
	} else if (c <= -1.04e-54) {
		tmp = t_0;
	} else if (c <= -2.2e-105) {
		tmp = t_1;
	} else if (c <= -1.1e-113) {
		tmp = t_2;
	} else if (c <= 2400.0) {
		tmp = t_0;
	} else if ((c <= 5.2e+26) || !(c <= 9.5e+99)) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -(a / d)
	t_1 = (b - (a * (d / c))) / c
	t_2 = (c / d) * (b / d)
	tmp = 0
	if c <= -2.05e+19:
		tmp = t_1
	elif c <= -1.04e-54:
		tmp = t_0
	elif c <= -2.2e-105:
		tmp = t_1
	elif c <= -1.1e-113:
		tmp = t_2
	elif c <= 2400.0:
		tmp = t_0
	elif (c <= 5.2e+26) or not (c <= 9.5e+99):
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(-Float64(a / d))
	t_1 = Float64(Float64(b - Float64(a * Float64(d / c))) / c)
	t_2 = Float64(Float64(c / d) * Float64(b / d))
	tmp = 0.0
	if (c <= -2.05e+19)
		tmp = t_1;
	elseif (c <= -1.04e-54)
		tmp = t_0;
	elseif (c <= -2.2e-105)
		tmp = t_1;
	elseif (c <= -1.1e-113)
		tmp = t_2;
	elseif (c <= 2400.0)
		tmp = t_0;
	elseif ((c <= 5.2e+26) || !(c <= 9.5e+99))
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -(a / d);
	t_1 = (b - (a * (d / c))) / c;
	t_2 = (c / d) * (b / d);
	tmp = 0.0;
	if (c <= -2.05e+19)
		tmp = t_1;
	elseif (c <= -1.04e-54)
		tmp = t_0;
	elseif (c <= -2.2e-105)
		tmp = t_1;
	elseif (c <= -1.1e-113)
		tmp = t_2;
	elseif (c <= 2400.0)
		tmp = t_0;
	elseif ((c <= 5.2e+26) || ~((c <= 9.5e+99)))
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = (-N[(a / d), $MachinePrecision])}, Block[{t$95$1 = N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.05e+19], t$95$1, If[LessEqual[c, -1.04e-54], t$95$0, If[LessEqual[c, -2.2e-105], t$95$1, If[LessEqual[c, -1.1e-113], t$95$2, If[LessEqual[c, 2400.0], t$95$0, If[Or[LessEqual[c, 5.2e+26], N[Not[LessEqual[c, 9.5e+99]], $MachinePrecision]], t$95$1, t$95$2]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.05e19 or -1.04e-54 < c < -2.20000000000000004e-105 or 2400 < c < 5.20000000000000004e26 or 9.49999999999999908e99 < c

   1. Initial program 46.4%

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

   2. Taylor expanded in c around inf 74.4%

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

      1. +-commutative 74.4%

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

      2. mul-1-neg 74.4%

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

      3. unsub-neg 74.4%

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

      4. unpow2 74.4%

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

      5. times-frac 82.4%

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

   4. Simplified 82.4%

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

      1. associate-*l/ 81.6%

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

      2. sub-div 81.6%

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

   6. Applied egg-rr 81.6%

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

   if -2.05e19 < c < -1.04e-54 or -1.10000000000000002e-113 < c < 2400

   1. Initial program 70.2%

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

   2. Taylor expanded in c around 0 75.9%

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

      1. associate-*r/ 75.9%

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

      2. neg-mul-1 75.9%

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

   4. Simplified 75.9%

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

   if -2.20000000000000004e-105 < c < -1.10000000000000002e-113 or 5.20000000000000004e26 < c < 9.49999999999999908e99

   1. Initial program 72.6%

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

      1. div-sub 72.6%

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

      2. *-commutative 72.6%

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

      3. add-sqr-sqrt 72.6%

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

      4. times-frac 86.4%

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

      5. fma-neg 86.4%

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

      6. hypot-def 86.4%

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

      7. hypot-def 94.9%

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

      8. associate-/l* 90.6%

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

      9. add-sqr-sqrt 90.6%

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

      10. pow2 90.6%

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

      11. hypot-def 90.6%

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

   3. Applied egg-rr 90.6%

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

   4. Taylor expanded in b around inf 57.3%

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

      1. *-commutative 57.3%

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

      2. unpow2 57.3%

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

      3. +-commutative 57.3%

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

      4. unpow2 57.3%

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

   6. Simplified 57.3%

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

   7. Taylor expanded in c around 0 52.9%

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

      1. *-commutative 52.9%

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

      2. unpow2 52.9%

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

      3. times-frac 70.7%

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

   9. Simplified 70.7%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.05 \cdot 10^{+19}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -1.04 \cdot 10^{-54}:\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{elif}\;c \leq -2.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{-113}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d}\\ \mathbf{elif}\;c \leq 2400:\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{+26} \lor \neg \left(c \leq 9.5 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d}\\ \end{array} \]

Alternative 6: 66.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{a}{d}\\ t_1 := \frac{b - d \cdot \frac{a}{c}}{c}\\ t_2 := \frac{c}{d} \cdot \frac{b}{d}\\ \mathbf{if}\;c \leq -5.6 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -8.7 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -1.25 \cdot 10^{-105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{-113}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 3100:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{+99}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (/ a d)))
        (t_1 (/ (- b (* d (/ a c))) c))
        (t_2 (* (/ c d) (/ b d))))
   (if (<= c -5.6e+18)
     t_1
     (if (<= c -8.7e-57)
       t_0
       (if (<= c -1.25e-105)
         t_1
         (if (<= c -1.1e-113)
           t_2
           (if (<= c 3100.0)
             t_0
             (if (<= c 3.2e+26)
               (/ (- b (* a (/ d c))) c)
               (if (<= c 9.5e+99) t_2 t_1)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -(a / d);
	double t_1 = (b - (d * (a / c))) / c;
	double t_2 = (c / d) * (b / d);
	double tmp;
	if (c <= -5.6e+18) {
		tmp = t_1;
	} else if (c <= -8.7e-57) {
		tmp = t_0;
	} else if (c <= -1.25e-105) {
		tmp = t_1;
	} else if (c <= -1.1e-113) {
		tmp = t_2;
	} else if (c <= 3100.0) {
		tmp = t_0;
	} else if (c <= 3.2e+26) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 9.5e+99) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_0 = -(a / d)
    t_1 = (b - (d * (a / c))) / c
    t_2 = (c / d) * (b / d)
    if (c <= (-5.6d+18)) then
        tmp = t_1
    else if (c <= (-8.7d-57)) then
        tmp = t_0
    else if (c <= (-1.25d-105)) then
        tmp = t_1
    else if (c <= (-1.1d-113)) then
        tmp = t_2
    else if (c <= 3100.0d0) then
        tmp = t_0
    else if (c <= 3.2d+26) then
        tmp = (b - (a * (d / c))) / c
    else if (c <= 9.5d+99) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = -(a / d);
	double t_1 = (b - (d * (a / c))) / c;
	double t_2 = (c / d) * (b / d);
	double tmp;
	if (c <= -5.6e+18) {
		tmp = t_1;
	} else if (c <= -8.7e-57) {
		tmp = t_0;
	} else if (c <= -1.25e-105) {
		tmp = t_1;
	} else if (c <= -1.1e-113) {
		tmp = t_2;
	} else if (c <= 3100.0) {
		tmp = t_0;
	} else if (c <= 3.2e+26) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 9.5e+99) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -(a / d)
	t_1 = (b - (d * (a / c))) / c
	t_2 = (c / d) * (b / d)
	tmp = 0
	if c <= -5.6e+18:
		tmp = t_1
	elif c <= -8.7e-57:
		tmp = t_0
	elif c <= -1.25e-105:
		tmp = t_1
	elif c <= -1.1e-113:
		tmp = t_2
	elif c <= 3100.0:
		tmp = t_0
	elif c <= 3.2e+26:
		tmp = (b - (a * (d / c))) / c
	elif c <= 9.5e+99:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(-Float64(a / d))
	t_1 = Float64(Float64(b - Float64(d * Float64(a / c))) / c)
	t_2 = Float64(Float64(c / d) * Float64(b / d))
	tmp = 0.0
	if (c <= -5.6e+18)
		tmp = t_1;
	elseif (c <= -8.7e-57)
		tmp = t_0;
	elseif (c <= -1.25e-105)
		tmp = t_1;
	elseif (c <= -1.1e-113)
		tmp = t_2;
	elseif (c <= 3100.0)
		tmp = t_0;
	elseif (c <= 3.2e+26)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= 9.5e+99)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -(a / d);
	t_1 = (b - (d * (a / c))) / c;
	t_2 = (c / d) * (b / d);
	tmp = 0.0;
	if (c <= -5.6e+18)
		tmp = t_1;
	elseif (c <= -8.7e-57)
		tmp = t_0;
	elseif (c <= -1.25e-105)
		tmp = t_1;
	elseif (c <= -1.1e-113)
		tmp = t_2;
	elseif (c <= 3100.0)
		tmp = t_0;
	elseif (c <= 3.2e+26)
		tmp = (b - (a * (d / c))) / c;
	elseif (c <= 9.5e+99)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = (-N[(a / d), $MachinePrecision])}, Block[{t$95$1 = N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -5.6e+18], t$95$1, If[LessEqual[c, -8.7e-57], t$95$0, If[LessEqual[c, -1.25e-105], t$95$1, If[LessEqual[c, -1.1e-113], t$95$2, If[LessEqual[c, 3100.0], t$95$0, If[LessEqual[c, 3.2e+26], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 9.5e+99], t$95$2, t$95$1]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -5.6e18 or -8.7000000000000002e-57 < c < -1.24999999999999991e-105 or 9.49999999999999908e99 < c

   1. Initial program 44.3%

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

   2. Taylor expanded in c around inf 73.5%

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

      1. +-commutative 73.5%

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

      2. mul-1-neg 73.5%

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

      3. unsub-neg 73.5%

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

      4. unpow2 73.5%

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

      5. times-frac 81.8%

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

   4. Simplified 81.8%

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

      1. associate-*r/ 82.9%

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

      2. sub-div 82.9%

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

   6. Applied egg-rr 82.9%

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

   if -5.6e18 < c < -8.7000000000000002e-57 or -1.10000000000000002e-113 < c < 3100

   1. Initial program 70.2%

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

   2. Taylor expanded in c around 0 75.9%

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

      1. associate-*r/ 75.9%

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

      2. neg-mul-1 75.9%

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

   4. Simplified 75.9%

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

   if -1.24999999999999991e-105 < c < -1.10000000000000002e-113 or 3.20000000000000029e26 < c < 9.49999999999999908e99

   1. Initial program 72.6%

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

      1. div-sub 72.6%

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

      2. *-commutative 72.6%

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

      3. add-sqr-sqrt 72.6%

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

      4. times-frac 86.4%

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

      5. fma-neg 86.4%

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

      6. hypot-def 86.4%

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

      7. hypot-def 94.9%

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

      8. associate-/l* 90.6%

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

      9. add-sqr-sqrt 90.6%

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

      10. pow2 90.6%

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

      11. hypot-def 90.6%

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

   3. Applied egg-rr 90.6%

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

   4. Taylor expanded in b around inf 57.3%

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

      1. *-commutative 57.3%

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

      2. unpow2 57.3%

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

      3. +-commutative 57.3%

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

      4. unpow2 57.3%

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

   6. Simplified 57.3%

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

   7. Taylor expanded in c around 0 52.9%

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

      1. *-commutative 52.9%

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

      2. unpow2 52.9%

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

      3. times-frac 70.7%

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

   9. Simplified 70.7%

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

   if 3100 < c < 3.20000000000000029e26

   1. Initial program 99.1%

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

   2. Taylor expanded in c around inf 95.2%

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

      1. +-commutative 95.2%

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

      2. mul-1-neg 95.2%

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

      3. unsub-neg 95.2%

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

      4. unpow2 95.2%

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

      5. times-frac 95.5%

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

   4. Simplified 95.5%

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

      1. associate-*l/ 95.5%

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

      2. sub-div 95.5%

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

   6. Applied egg-rr 95.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -8.7 \cdot 10^{-57}:\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{elif}\;c \leq -1.25 \cdot 10^{-105}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{-113}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d}\\ \mathbf{elif}\;c \leq 3100:\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]

Alternative 7: 79.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{if}\;c \leq -8.6 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 1.1 \cdot 10^{+102}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* d (/ a c))) c)))
   (if (<= c -8.6e+52)
     t_0
     (if (<= c 2.5e-89)
       (- (* (/ c d) (/ b d)) (/ a d))
       (if (<= c 1.1e+102) (/ (- (* c b) (* d a)) (+ (* c c) (* d d))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - (d * (a / c))) / c;
	double tmp;
	if (c <= -8.6e+52) {
		tmp = t_0;
	} else if (c <= 2.5e-89) {
		tmp = ((c / d) * (b / d)) - (a / d);
	} else if (c <= 1.1e+102) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b - (d * (a / c))) / c
    if (c <= (-8.6d+52)) then
        tmp = t_0
    else if (c <= 2.5d-89) then
        tmp = ((c / d) * (b / d)) - (a / d)
    else if (c <= 1.1d+102) then
        tmp = ((c * b) - (d * a)) / ((c * c) + (d * d))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b - (d * (a / c))) / c;
	double tmp;
	if (c <= -8.6e+52) {
		tmp = t_0;
	} else if (c <= 2.5e-89) {
		tmp = ((c / d) * (b / d)) - (a / d);
	} else if (c <= 1.1e+102) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b - (d * (a / c))) / c
	tmp = 0
	if c <= -8.6e+52:
		tmp = t_0
	elif c <= 2.5e-89:
		tmp = ((c / d) * (b / d)) - (a / d)
	elif c <= 1.1e+102:
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(d * Float64(a / c))) / c)
	tmp = 0.0
	if (c <= -8.6e+52)
		tmp = t_0;
	elseif (c <= 2.5e-89)
		tmp = Float64(Float64(Float64(c / d) * Float64(b / d)) - Float64(a / d));
	elseif (c <= 1.1e+102)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b - (d * (a / c))) / c;
	tmp = 0.0;
	if (c <= -8.6e+52)
		tmp = t_0;
	elseif (c <= 2.5e-89)
		tmp = ((c / d) * (b / d)) - (a / d);
	elseif (c <= 1.1e+102)
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -8.6e+52], t$95$0, If[LessEqual[c, 2.5e-89], N[(N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.1e+102], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -8.5999999999999999e52 or 1.10000000000000004e102 < c

   1. Initial program 39.7%

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

   2. Taylor expanded in c around inf 76.0%

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

      1. +-commutative 76.0%

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

      2. mul-1-neg 76.0%

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

      3. unsub-neg 76.0%

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

      4. unpow2 76.0%

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

      5. times-frac 85.5%

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

   4. Simplified 85.5%

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

      1. associate-*r/ 86.7%

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

      2. sub-div 86.7%

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

   6. Applied egg-rr 86.7%

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

   if -8.5999999999999999e52 < c < 2.49999999999999983e-89

   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 77.3%

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

      1. +-commutative 77.3%

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

      2. mul-1-neg 77.3%

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

      3. unsub-neg 77.3%

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

      4. *-commutative 77.3%

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

      5. unpow2 77.3%

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

      6. times-frac 82.0%

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

   4. Simplified 82.0%

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

   if 2.49999999999999983e-89 < c < 1.10000000000000004e102

   1. Initial program 80.1%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8.6 \cdot 10^{+52}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 1.1 \cdot 10^{+102}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]

Alternative 8: 76.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ t_1 := \frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{if}\;c \leq -5.2 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 19000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{+23}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 1.35 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* (/ c d) (/ b d)) (/ a d))) (t_1 (/ (- b (* d (/ a c))) c)))
   (if (<= c -5.2e+53)
     t_1
     (if (<= c 19000000000.0)
       t_0
       (if (<= c 2.3e+23)
         (/ (- b (* a (/ d c))) c)
         (if (<= c 1.35e+90) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c / d) * (b / d)) - (a / d);
	double t_1 = (b - (d * (a / c))) / c;
	double tmp;
	if (c <= -5.2e+53) {
		tmp = t_1;
	} else if (c <= 19000000000.0) {
		tmp = t_0;
	} else if (c <= 2.3e+23) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 1.35e+90) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((c / d) * (b / d)) - (a / d)
    t_1 = (b - (d * (a / c))) / c
    if (c <= (-5.2d+53)) then
        tmp = t_1
    else if (c <= 19000000000.0d0) then
        tmp = t_0
    else if (c <= 2.3d+23) then
        tmp = (b - (a * (d / c))) / c
    else if (c <= 1.35d+90) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c / d) * (b / d)) - (a / d);
	double t_1 = (b - (d * (a / c))) / c;
	double tmp;
	if (c <= -5.2e+53) {
		tmp = t_1;
	} else if (c <= 19000000000.0) {
		tmp = t_0;
	} else if (c <= 2.3e+23) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 1.35e+90) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c / d) * (b / d)) - (a / d)
	t_1 = (b - (d * (a / c))) / c
	tmp = 0
	if c <= -5.2e+53:
		tmp = t_1
	elif c <= 19000000000.0:
		tmp = t_0
	elif c <= 2.3e+23:
		tmp = (b - (a * (d / c))) / c
	elif c <= 1.35e+90:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c / d) * Float64(b / d)) - Float64(a / d))
	t_1 = Float64(Float64(b - Float64(d * Float64(a / c))) / c)
	tmp = 0.0
	if (c <= -5.2e+53)
		tmp = t_1;
	elseif (c <= 19000000000.0)
		tmp = t_0;
	elseif (c <= 2.3e+23)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= 1.35e+90)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c / d) * (b / d)) - (a / d);
	t_1 = (b - (d * (a / c))) / c;
	tmp = 0.0;
	if (c <= -5.2e+53)
		tmp = t_1;
	elseif (c <= 19000000000.0)
		tmp = t_0;
	elseif (c <= 2.3e+23)
		tmp = (b - (a * (d / c))) / c;
	elseif (c <= 1.35e+90)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -5.2e+53], t$95$1, If[LessEqual[c, 19000000000.0], t$95$0, If[LessEqual[c, 2.3e+23], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 1.35e+90], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -5.19999999999999996e53 or 1.35e90 < c

   1. Initial program 41.9%

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

   2. Taylor expanded in c around inf 75.2%

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

      1. +-commutative 75.2%

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

      2. mul-1-neg 75.2%

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

      3. unsub-neg 75.2%

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

      4. unpow2 75.2%

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

      5. times-frac 84.2%

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

   4. Simplified 84.2%

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

      1. associate-*r/ 85.5%

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

      2. sub-div 85.5%

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

   6. Applied egg-rr 85.5%

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

   if -5.19999999999999996e53 < c < 1.9e10 or 2.3e23 < c < 1.35e90

   1. Initial program 70.4%

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

   2. Taylor expanded in c around 0 73.9%

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

      1. +-commutative 73.9%

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

      2. mul-1-neg 73.9%

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

      3. unsub-neg 73.9%

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

      4. *-commutative 73.9%

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

      5. unpow2 73.9%

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

      6. times-frac 81.2%

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

   4. Simplified 81.2%

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

   if 1.9e10 < c < 2.3e23

   1. Initial program 99.1%

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

   2. Taylor expanded in c around inf 95.2%

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

      1. +-commutative 95.2%

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

      2. mul-1-neg 95.2%

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

      3. unsub-neg 95.2%

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

      4. unpow2 95.2%

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

      5. times-frac 95.5%

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

   4. Simplified 95.5%

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

      1. associate-*l/ 95.5%

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

      2. sub-div 95.5%

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

   6. Applied egg-rr 95.5%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 19000000000:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{+23}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 1.35 \cdot 10^{+90}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]

Alternative 9: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{if}\;c \leq -7 \cdot 10^{+55}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 6500000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 6.6 \cdot 10^{+92}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c \cdot \frac{c}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* (/ c d) (/ b d)) (/ a d))))
   (if (<= c -7e+55)
     (/ (- b (* d (/ a c))) c)
     (if (<= c 6500000000.0)
       t_0
       (if (<= c 8.5e+19)
         (/ (- b (* a (/ d c))) c)
         (if (<= c 6.6e+92) t_0 (- (/ b c) (/ d (* c (/ c a))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c / d) * (b / d)) - (a / d);
	double tmp;
	if (c <= -7e+55) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= 6500000000.0) {
		tmp = t_0;
	} else if (c <= 8.5e+19) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 6.6e+92) {
		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) :: tmp
    t_0 = ((c / d) * (b / d)) - (a / d)
    if (c <= (-7d+55)) then
        tmp = (b - (d * (a / c))) / c
    else if (c <= 6500000000.0d0) then
        tmp = t_0
    else if (c <= 8.5d+19) then
        tmp = (b - (a * (d / c))) / c
    else if (c <= 6.6d+92) 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 / d) * (b / d)) - (a / d);
	double tmp;
	if (c <= -7e+55) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= 6500000000.0) {
		tmp = t_0;
	} else if (c <= 8.5e+19) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 6.6e+92) {
		tmp = t_0;
	} else {
		tmp = (b / c) - (d / (c * (c / a)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c / d) * (b / d)) - (a / d)
	tmp = 0
	if c <= -7e+55:
		tmp = (b - (d * (a / c))) / c
	elif c <= 6500000000.0:
		tmp = t_0
	elif c <= 8.5e+19:
		tmp = (b - (a * (d / c))) / c
	elif c <= 6.6e+92:
		tmp = t_0
	else:
		tmp = (b / c) - (d / (c * (c / a)))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c / d) * Float64(b / d)) - Float64(a / d))
	tmp = 0.0
	if (c <= -7e+55)
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	elseif (c <= 6500000000.0)
		tmp = t_0;
	elseif (c <= 8.5e+19)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= 6.6e+92)
		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 / d) * (b / d)) - (a / d);
	tmp = 0.0;
	if (c <= -7e+55)
		tmp = (b - (d * (a / c))) / c;
	elseif (c <= 6500000000.0)
		tmp = t_0;
	elseif (c <= 8.5e+19)
		tmp = (b - (a * (d / c))) / c;
	elseif (c <= 6.6e+92)
		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[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -7e+55], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 6500000000.0], t$95$0, If[LessEqual[c, 8.5e+19], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 6.6e+92], 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 < -7.00000000000000021e55

   1. Initial program 49.9%

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

   2. Taylor expanded in c around inf 79.1%

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

      1. +-commutative 79.1%

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

      2. mul-1-neg 79.1%

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

      3. unsub-neg 79.1%

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

      4. unpow2 79.1%

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

      5. times-frac 88.5%

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

   4. Simplified 88.5%

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

      1. associate-*r/ 90.0%

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

      2. sub-div 90.0%

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

   6. Applied egg-rr 90.0%

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

   if -7.00000000000000021e55 < c < 6.5e9 or 8.5e19 < c < 6.59999999999999948e92

   1. Initial program 70.4%

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

   2. Taylor expanded in c around 0 73.9%

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

      1. +-commutative 73.9%

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

      2. mul-1-neg 73.9%

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

      3. unsub-neg 73.9%

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

      4. *-commutative 73.9%

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

      5. unpow2 73.9%

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

      6. times-frac 81.2%

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

   4. Simplified 81.2%

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

   if 6.5e9 < c < 8.5e19

   1. Initial program 99.1%

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

   2. Taylor expanded in c around inf 95.2%

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

      1. +-commutative 95.2%

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

      2. mul-1-neg 95.2%

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

      3. unsub-neg 95.2%

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

      4. unpow2 95.2%

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

      5. times-frac 95.5%

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

   4. Simplified 95.5%

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

      1. associate-*l/ 95.5%

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

      2. sub-div 95.5%

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

   6. Applied egg-rr 95.5%

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

   if 6.59999999999999948e92 < c

   1. Initial program 31.6%

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

   2. Taylor expanded in c around inf 70.2%

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

      1. +-commutative 70.2%

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

      2. mul-1-neg 70.2%

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

      3. unsub-neg 70.2%

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

      4. unpow2 70.2%

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

      5. times-frac 78.8%

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

   4. Simplified 78.8%

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

      1. clear-num 78.9%

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

      2. frac-times 79.8%

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

      3. *-un-lft-identity 79.8%

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

   6. Applied egg-rr 79.8%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7 \cdot 10^{+55}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 6500000000:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 6.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c \cdot \frac{c}{a}}\\ \end{array} \]

Alternative 10: 61.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-30}:\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -7.8e+18)
   (/ b c)
   (if (<= c 6.5e-30)
     (- (/ a d))
     (if (<= c 9.5e+99) (* (/ c d) (/ b d)) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7.8e+18) {
		tmp = b / c;
	} else if (c <= 6.5e-30) {
		tmp = -(a / d);
	} else if (c <= 9.5e+99) {
		tmp = (c / d) * (b / 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 (c <= (-7.8d+18)) then
        tmp = b / c
    else if (c <= 6.5d-30) then
        tmp = -(a / d)
    else if (c <= 9.5d+99) then
        tmp = (c / d) * (b / 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 (c <= -7.8e+18) {
		tmp = b / c;
	} else if (c <= 6.5e-30) {
		tmp = -(a / d);
	} else if (c <= 9.5e+99) {
		tmp = (c / d) * (b / d);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -7.8e+18:
		tmp = b / c
	elif c <= 6.5e-30:
		tmp = -(a / d)
	elif c <= 9.5e+99:
		tmp = (c / d) * (b / d)
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -7.8e+18)
		tmp = Float64(b / c);
	elseif (c <= 6.5e-30)
		tmp = Float64(-Float64(a / d));
	elseif (c <= 9.5e+99)
		tmp = Float64(Float64(c / d) * Float64(b / d));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -7.8e+18)
		tmp = b / c;
	elseif (c <= 6.5e-30)
		tmp = -(a / d);
	elseif (c <= 9.5e+99)
		tmp = (c / d) * (b / d);
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -7.8e+18], N[(b / c), $MachinePrecision], If[LessEqual[c, 6.5e-30], (-N[(a / d), $MachinePrecision]), If[LessEqual[c, 9.5e+99], N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.8e18 or 9.49999999999999908e99 < c

   1. Initial program 42.8%

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

   2. Taylor expanded in c around inf 66.1%

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

   if -7.8e18 < c < 6.5000000000000005e-30

   1. Initial program 71.7%

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

   2. Taylor expanded in c around 0 72.4%

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

      1. associate-*r/ 72.4%

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

      2. neg-mul-1 72.4%

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

   4. Simplified 72.4%

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

   if 6.5000000000000005e-30 < c < 9.49999999999999908e99

   1. Initial program 71.9%

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

      1. div-sub 71.9%

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

      2. *-commutative 71.9%

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

      3. add-sqr-sqrt 71.8%

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

      4. times-frac 83.9%

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

      5. fma-neg 83.9%

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

      6. hypot-def 83.9%

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

      7. hypot-def 95.5%

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

      8. associate-/l* 91.7%

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

      9. add-sqr-sqrt 91.7%

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

      10. pow2 91.7%

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

      11. hypot-def 91.7%

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

   3. Applied egg-rr 91.7%

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

   4. Taylor expanded in b around inf 48.1%

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

      1. *-commutative 48.1%

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

      2. unpow2 48.1%

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

      3. +-commutative 48.1%

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

      4. unpow2 48.1%

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

   6. Simplified 48.1%

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

   7. Taylor expanded in c around 0 32.5%

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

      1. *-commutative 32.5%

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

      2. unpow2 32.5%

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

      3. times-frac 52.1%

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

   9. Simplified 52.1%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-30}:\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 11: 63.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.35 \cdot 10^{+18}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{+99}:\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.35e+18) (/ b c) (if (<= c 1.15e+99) (- (/ a d)) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.35e+18) {
		tmp = b / c;
	} else if (c <= 1.15e+99) {
		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 (c <= (-1.35d+18)) then
        tmp = b / c
    else if (c <= 1.15d+99) 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 (c <= -1.35e+18) {
		tmp = b / c;
	} else if (c <= 1.15e+99) {
		tmp = -(a / d);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.35e+18:
		tmp = b / c
	elif c <= 1.15e+99:
		tmp = -(a / d)
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.35e+18)
		tmp = Float64(b / c);
	elseif (c <= 1.15e+99)
		tmp = Float64(-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 (c <= -1.35e+18)
		tmp = b / c;
	elseif (c <= 1.15e+99)
		tmp = -(a / d);
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.35e+18], N[(b / c), $MachinePrecision], If[LessEqual[c, 1.15e+99], (-N[(a / d), $MachinePrecision]), N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.35e18 or 1.1500000000000001e99 < c

   1. Initial program 43.3%

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

   2. Taylor expanded in c around inf 65.6%

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

   if -1.35e18 < c < 1.1500000000000001e99

   1. Initial program 71.5%

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

   2. Taylor expanded in c around 0 66.0%

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

      1. associate-*r/ 66.0%

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

      2. neg-mul-1 66.0%

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

   4. Simplified 66.0%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.35 \cdot 10^{+18}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{+99}:\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 12: 42.5% 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 58.6%

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

2. Taylor expanded in c around inf 39.7%

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

3. Final simplification 39.7%

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

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 2023293 
(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))))