Complex division, imag part

Percentage Accurate: 61.1% → 98.3%

Time: 14.8s

Alternatives: 12

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.1% 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: 98.3% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(a, b, c, d):
	return (((c / math.hypot(c, d)) * b) - (d * (a / math.hypot(c, d)))) / math.hypot(c, d)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.5%

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

   1. div-sub 61.7%

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

   2. sub-neg 61.7%

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

   3. *-un-lft-identity 61.7%

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

   4. add-sqr-sqrt 61.7%

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

   5. times-frac 61.7%

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

   6. fma-def 61.7%

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

   7. hypot-def 61.7%

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

   8. hypot-def 67.5%

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

   9. associate-/l* 70.6%

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

   10. add-sqr-sqrt 70.6%

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

   11. pow2 70.6%

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

   12. hypot-def 70.6%

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

4. Applied egg-rr 70.6%

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

   1. fma-neg 69.8%

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

   2. *-commutative 69.8%

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

   3. associate-/l* 79.4%

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

   4. associate-/r/ 78.1%

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

   5. *-commutative 78.1%

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

6. Simplified 78.1%

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

   1. *-un-lft-identity 78.1%

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

   2. unpow2 78.1%

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

   3. times-frac 85.6%

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

8. Applied egg-rr 85.6%

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

   1. associate-*l/ 85.6%

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

   2. *-lft-identity 85.6%

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

10. Simplified 85.6%

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

    1. associate-*l/ 85.7%

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

    2. *-un-lft-identity 85.7%

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

    3. associate-*r/ 96.6%

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

    4. sub-div 97.0%

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

    5. associate-/r/ 99.2%

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

12. Applied egg-rr 99.2%

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

13. Final simplification 99.2%

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

Alternative 2: 88.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot b - d \cdot a\\ \mathbf{if}\;\frac{t_0}{c \cdot c + d \cdot d} \leq 10^{+254}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c b) (* d a))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 1e+254)
     (* (/ 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+254) {
		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+254)
		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+254], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[((-a) / d), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 9.9999999999999994e253

   1. Initial program 82.7%

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

      1. *-un-lft-identity 82.7%

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

      2. add-sqr-sqrt 82.7%

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

      3. times-frac 82.7%

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

      4. hypot-def 82.7%

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

      5. hypot-def 97.0%

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

   4. Applied egg-rr 97.0%

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

   if 9.9999999999999994e253 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 11.4%

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

      1. div-sub 9.4%

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

      2. sub-neg 9.4%

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

      3. *-commutative 9.4%

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

      4. add-sqr-sqrt 9.4%

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

      5. times-frac 16.3%

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

      6. fma-def 16.3%

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

      7. hypot-def 16.3%

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

      8. hypot-def 45.6%

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

      9. associate-/l* 58.1%

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

      10. add-sqr-sqrt 58.1%

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

      11. pow2 58.1%

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

      12. hypot-def 58.1%

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

   4. Applied egg-rr 58.1%

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

   5. Taylor expanded in c around 0 67.8%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 10^{+254}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.55 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{d}{c} \cdot \frac{a}{c}, \frac{b}{c}\right)\\ \mathbf{elif}\;c \leq -4.2 \cdot 10^{-100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{-145}:\\ \;\;\;\;\frac{1}{d} \cdot \left(c \cdot \frac{b}{d} - a\right)\\ \mathbf{elif}\;c \leq 3.9 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.2 \cdot 10^{+41}:\\ \;\;\;\;c \cdot \frac{\frac{b}{d}}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{\mathsf{hypot}\left(c, d\right)}{a}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (<= c -1.55e+95)
     (fma -1.0 (* (/ d c) (/ a c)) (/ b c))
     (if (<= c -4.2e-100)
       t_0
       (if (<= c 3.1e-145)
         (* (/ 1.0 d) (- (* c (/ b d)) a))
         (if (<= c 3.9e-66)
           t_0
           (if (<= c 1.2e+41)
             (- (* c (/ (/ b d) d)) (/ a d))
             (/ (- b (/ d (/ (hypot c d) a))) (hypot c d)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1.55e+95) {
		tmp = fma(-1.0, ((d / c) * (a / c)), (b / c));
	} else if (c <= -4.2e-100) {
		tmp = t_0;
	} else if (c <= 3.1e-145) {
		tmp = (1.0 / d) * ((c * (b / d)) - a);
	} else if (c <= 3.9e-66) {
		tmp = t_0;
	} else if (c <= 1.2e+41) {
		tmp = (c * ((b / d) / d)) - (a / d);
	} else {
		tmp = (b - (d / (hypot(c, d) / a))) / hypot(c, d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -1.55e+95)
		tmp = fma(-1.0, Float64(Float64(d / c) * Float64(a / c)), Float64(b / c));
	elseif (c <= -4.2e-100)
		tmp = t_0;
	elseif (c <= 3.1e-145)
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(c * Float64(b / d)) - a));
	elseif (c <= 3.9e-66)
		tmp = t_0;
	elseif (c <= 1.2e+41)
		tmp = Float64(Float64(c * Float64(Float64(b / d) / d)) - Float64(a / d));
	else
		tmp = Float64(Float64(b - Float64(d / Float64(hypot(c, d) / a))) / hypot(c, d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.55e+95], N[(-1.0 * N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -4.2e-100], t$95$0, If[LessEqual[c, 3.1e-145], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.9e-66], t$95$0, If[LessEqual[c, 1.2e+41], N[(N[(c * N[(N[(b / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], N[(N[(b - N[(d / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -1.5500000000000001e95

   1. Initial program 49.8%

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

   3. Taylor expanded in c around inf 84.3%

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

      1. fma-def 84.3%

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

      2. associate-/l* 84.9%

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

   5. Simplified 84.9%

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

      1. clear-num 84.9%

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

      2. inv-pow 84.9%

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

   7. Applied egg-rr 84.9%

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

      1. unpow-1 84.9%

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

      2. associate-/l/ 84.3%

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

      3. *-commutative 84.3%

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

   9. Simplified 84.3%

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

       1. clear-num 84.3%

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

       2. unpow2 84.3%

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

       3. times-frac 94.8%

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

   11. Applied egg-rr 94.8%

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

   if -1.5500000000000001e95 < c < -4.20000000000000019e-100 or 3.1e-145 < c < 3.89999999999999983e-66

   1. Initial program 93.2%

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

   if -4.20000000000000019e-100 < c < 3.1e-145

   1. Initial program 59.3%

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

   3. Taylor expanded in c around 0 76.6%

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

      1. +-commutative 76.6%

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

      2. mul-1-neg 76.6%

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

      3. unsub-neg 76.6%

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

      4. associate-/l* 76.8%

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

      5. associate-/r/ 71.0%

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

   5. Simplified 71.0%

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

      1. *-un-lft-identity 71.0%

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

      2. unpow2 71.0%

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

      3. times-frac 74.5%

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

   7. Applied egg-rr 74.5%

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

      1. associate-*l* 81.6%

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

      2. fma-neg 81.6%

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

   9. Applied egg-rr 81.6%

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

       1. fma-udef 81.6%

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

       2. *-lft-identity 81.6%

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

       3. associate-*l/ 81.3%

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

       4. distribute-rgt-neg-in 81.3%

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

       5. distribute-lft-out 81.3%

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

       6. *-commutative 81.3%

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

   11. Simplified 81.3%

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

   if 3.89999999999999983e-66 < c < 1.2000000000000001e41

   1. Initial program 62.6%

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

   3. Taylor expanded in c around 0 68.2%

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

      1. +-commutative 68.2%

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

      2. mul-1-neg 68.2%

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

      3. unsub-neg 68.2%

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

      4. associate-/l* 68.2%

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

      5. associate-/r/ 68.1%

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

   5. Simplified 68.1%

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

      1. *-un-lft-identity 68.1%

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

      2. unpow2 68.1%

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

      3. times-frac 76.3%

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

   7. Applied egg-rr 76.3%

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

      1. associate-*l/ 76.3%

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

      2. *-lft-identity 76.3%

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

   9. Simplified 76.3%

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

   if 1.2000000000000001e41 < c

   1. Initial program 43.5%

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

      1. div-sub 43.5%

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

      2. sub-neg 43.5%

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

      3. *-un-lft-identity 43.5%

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

      4. add-sqr-sqrt 43.5%

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

      5. times-frac 43.4%

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

      6. fma-def 43.4%

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

      7. hypot-def 43.4%

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

      8. hypot-def 54.9%

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

      9. associate-/l* 52.7%

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

      10. add-sqr-sqrt 52.7%

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

      11. pow2 52.7%

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

      12. hypot-def 52.7%

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

   4. Applied egg-rr 52.7%

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

      1. fma-neg 52.7%

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

      2. *-commutative 52.7%

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

      3. associate-/l* 84.3%

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

      4. associate-/r/ 86.8%

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

      5. *-commutative 86.8%

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

   6. Simplified 86.8%

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

   7. Taylor expanded in c around -inf 28.6%

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

      1. mul-1-neg 28.6%

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

   9. Simplified 28.6%

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

       1. associate-*l/ 28.6%

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

       2. *-un-lft-identity 28.6%

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

       3. associate-*r/ 28.2%

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

       4. unpow2 28.2%

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

       5. frac-times 36.2%

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

       6. associate-*l/ 38.1%

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

       7. sub-div 38.1%

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

       8. add-sqr-sqrt 22.6%

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

       9. sqrt-unprod 55.3%

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

       10. sqr-neg 55.3%

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

       11. sqrt-unprod 46.4%

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

       12. add-sqr-sqrt 92.5%

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

       13. clear-num 92.5%

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

       14. un-div-inv 92.5%

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

   11. Applied egg-rr 92.5%

       \[\leadsto \color{blue}{\frac{b - \frac{d}{\frac{\mathsf{hypot}\left(c, d\right)}{a}}}{\mathsf{hypot}\left(c, d\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.55 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{d}{c} \cdot \frac{a}{c}, \frac{b}{c}\right)\\ \mathbf{elif}\;c \leq -4.2 \cdot 10^{-100}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{-145}:\\ \;\;\;\;\frac{1}{d} \cdot \left(c \cdot \frac{b}{d} - a\right)\\ \mathbf{elif}\;c \leq 3.9 \cdot 10^{-66}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.2 \cdot 10^{+41}:\\ \;\;\;\;c \cdot \frac{\frac{b}{d}}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{\mathsf{hypot}\left(c, d\right)}{a}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot b - d \cdot a\\ \mathbf{if}\;\frac{t_0}{c \cdot c + d \cdot d} \leq 10^{+254}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{\mathsf{hypot}\left(c, d\right)}{a}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c b) (* d a))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 1e+254)
     (* (/ 1.0 (hypot c d)) (/ t_0 (hypot c d)))
     (/ (- b (/ d (/ (hypot c d) a))) (hypot c 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+254) {
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	} else {
		tmp = (b - (d / (hypot(c, d) / a))) / hypot(c, d);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (c * b) - (d * a);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+254) {
		tmp = (1.0 / Math.hypot(c, d)) * (t_0 / Math.hypot(c, d));
	} else {
		tmp = (b - (d / (Math.hypot(c, d) / a))) / Math.hypot(c, d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (c * b) - (d * a)
	tmp = 0
	if (t_0 / ((c * c) + (d * d))) <= 1e+254:
		tmp = (1.0 / math.hypot(c, d)) * (t_0 / math.hypot(c, d))
	else:
		tmp = (b - (d / (math.hypot(c, d) / a))) / math.hypot(c, 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+254)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, d)));
	else
		tmp = Float64(Float64(b - Float64(d / Float64(hypot(c, d) / a))) / hypot(c, d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (c * b) - (d * a);
	tmp = 0.0;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+254)
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	else
		tmp = (b - (d / (hypot(c, d) / a))) / hypot(c, d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+254], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - N[(d / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 9.9999999999999994e253

   1. Initial program 82.7%

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

      1. *-un-lft-identity 82.7%

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

      2. add-sqr-sqrt 82.7%

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

      3. times-frac 82.7%

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

      4. hypot-def 82.7%

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

      5. hypot-def 97.0%

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

   4. Applied egg-rr 97.0%

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

   if 9.9999999999999994e253 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 11.4%

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

      1. div-sub 9.4%

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

      2. sub-neg 9.4%

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

      3. *-un-lft-identity 9.4%

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

      4. add-sqr-sqrt 9.4%

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

      5. times-frac 9.4%

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

      6. fma-def 9.4%

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

      7. hypot-def 9.4%

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

      8. hypot-def 11.9%

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

      9. associate-/l* 20.1%

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

      10. add-sqr-sqrt 20.1%

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

      11. pow2 20.1%

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

      12. hypot-def 20.1%

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

   4. Applied egg-rr 20.1%

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

      1. fma-neg 18.6%

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

      2. *-commutative 18.6%

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

      3. associate-/l* 57.9%

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

      4. associate-/r/ 57.9%

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

      5. *-commutative 57.9%

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

   6. Simplified 57.9%

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

   7. Taylor expanded in c around -inf 29.7%

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

      1. mul-1-neg 29.7%

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

   9. Simplified 29.7%

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

       1. associate-*l/ 29.8%

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

       2. *-un-lft-identity 29.8%

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

       3. associate-*r/ 20.1%

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

       4. unpow2 20.1%

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

       5. frac-times 58.9%

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

       6. associate-*l/ 58.9%

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

       7. sub-div 60.3%

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

       8. add-sqr-sqrt 32.3%

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

       9. sqrt-unprod 52.4%

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

       10. sqr-neg 52.4%

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

       11. sqrt-unprod 37.1%

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

       12. add-sqr-sqrt 66.4%

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

       13. clear-num 66.3%

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

       14. un-div-inv 66.3%

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

   11. Applied egg-rr 66.3%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 10^{+254}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{\mathsf{hypot}\left(c, d\right)}{a}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := c \cdot b - d \cdot a\\ \mathbf{if}\;\frac{t_1}{c \cdot c + d \cdot d} \leq 10^{+254}:\\ \;\;\;\;t_0 \cdot \frac{t_1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{c}{\frac{\mathsf{hypot}\left(c, d\right)}{b}} - \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot c d))) (t_1 (- (* c b) (* d a))))
   (if (<= (/ t_1 (+ (* c c) (* d d))) 1e+254)
     (* t_0 (/ t_1 (hypot c d)))
     (- (* t_0 (/ c (/ (hypot c d) b))) (/ a d)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = 1.0 / hypot(c, d);
	double t_1 = (c * b) - (d * a);
	double tmp;
	if ((t_1 / ((c * c) + (d * d))) <= 1e+254) {
		tmp = t_0 * (t_1 / hypot(c, d));
	} else {
		tmp = (t_0 * (c / (hypot(c, d) / b))) - (a / d);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = 1.0 / Math.hypot(c, d);
	double t_1 = (c * b) - (d * a);
	double tmp;
	if ((t_1 / ((c * c) + (d * d))) <= 1e+254) {
		tmp = t_0 * (t_1 / Math.hypot(c, d));
	} else {
		tmp = (t_0 * (c / (Math.hypot(c, d) / b))) - (a / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = 1.0 / math.hypot(c, d)
	t_1 = (c * b) - (d * a)
	tmp = 0
	if (t_1 / ((c * c) + (d * d))) <= 1e+254:
		tmp = t_0 * (t_1 / math.hypot(c, d))
	else:
		tmp = (t_0 * (c / (math.hypot(c, d) / b))) - (a / d)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(1.0 / hypot(c, d))
	t_1 = Float64(Float64(c * b) - Float64(d * a))
	tmp = 0.0
	if (Float64(t_1 / Float64(Float64(c * c) + Float64(d * d))) <= 1e+254)
		tmp = Float64(t_0 * Float64(t_1 / hypot(c, d)));
	else
		tmp = Float64(Float64(t_0 * Float64(c / Float64(hypot(c, d) / b))) - Float64(a / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = 1.0 / hypot(c, d);
	t_1 = (c * b) - (d * a);
	tmp = 0.0;
	if ((t_1 / ((c * c) + (d * d))) <= 1e+254)
		tmp = t_0 * (t_1 / hypot(c, d));
	else
		tmp = (t_0 * (c / (hypot(c, d) / b))) - (a / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+254], N[(t$95$0 * N[(t$95$1 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(c / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 9.9999999999999994e253

   1. Initial program 82.7%

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

      1. *-un-lft-identity 82.7%

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

      2. add-sqr-sqrt 82.7%

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

      3. times-frac 82.7%

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

      4. hypot-def 82.7%

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

      5. hypot-def 97.0%

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

   4. Applied egg-rr 97.0%

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

   if 9.9999999999999994e253 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 11.4%

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

      1. div-sub 9.4%

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

      2. sub-neg 9.4%

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

      3. *-un-lft-identity 9.4%

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

      4. add-sqr-sqrt 9.4%

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

      5. times-frac 9.4%

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

      6. fma-def 9.4%

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

      7. hypot-def 9.4%

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

      8. hypot-def 11.9%

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

      9. associate-/l* 20.1%

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

      10. add-sqr-sqrt 20.1%

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

      11. pow2 20.1%

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

      12. hypot-def 20.1%

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

   4. Applied egg-rr 20.1%

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

      1. fma-neg 18.6%

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

      2. *-commutative 18.6%

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

      3. associate-/l* 57.9%

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

      4. associate-/r/ 57.9%

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

      5. *-commutative 57.9%

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

   6. Simplified 57.9%

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

      1. *-un-lft-identity 57.9%

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

      2. unpow2 57.9%

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

      3. times-frac 81.7%

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

   8. Applied egg-rr 81.7%

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

      1. associate-*l/ 81.8%

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

      2. *-lft-identity 81.8%

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

   10. Simplified 81.8%

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

   11. Taylor expanded in d around inf 67.8%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 10^{+254}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\frac{\mathsf{hypot}\left(c, d\right)}{b}} - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ t_1 := \mathsf{fma}\left(-1, \frac{d}{c} \cdot \frac{a}{c}, \frac{b}{c}\right)\\ \mathbf{if}\;c \leq -4 \cdot 10^{+91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -5 \cdot 10^{-100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.12 \cdot 10^{-142}:\\ \;\;\;\;\frac{1}{d} \cdot \left(c \cdot \frac{b}{d} - a\right)\\ \mathbf{elif}\;c \leq 1.45 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d))))
        (t_1 (fma -1.0 (* (/ d c) (/ a c)) (/ b c))))
   (if (<= c -4e+91)
     t_1
     (if (<= c -5e-100)
       t_0
       (if (<= c 1.12e-142)
         (* (/ 1.0 d) (- (* c (/ b d)) a))
         (if (<= c 1.45e+81) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double t_1 = fma(-1.0, ((d / c) * (a / c)), (b / c));
	double tmp;
	if (c <= -4e+91) {
		tmp = t_1;
	} else if (c <= -5e-100) {
		tmp = t_0;
	} else if (c <= 1.12e-142) {
		tmp = (1.0 / d) * ((c * (b / d)) - a);
	} else if (c <= 1.45e+81) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = fma(-1.0, Float64(Float64(d / c) * Float64(a / c)), Float64(b / c))
	tmp = 0.0
	if (c <= -4e+91)
		tmp = t_1;
	elseif (c <= -5e-100)
		tmp = t_0;
	elseif (c <= 1.12e-142)
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(c * Float64(b / d)) - a));
	elseif (c <= 1.45e+81)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 * N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -4e+91], t$95$1, If[LessEqual[c, -5e-100], t$95$0, If[LessEqual[c, 1.12e-142], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.45e+81], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -4.00000000000000032e91 or 1.45e81 < c

   1. Initial program 40.0%

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

   3. Taylor expanded in c around inf 80.9%

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

      1. fma-def 80.9%

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

      2. associate-/l* 82.3%

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

   5. Simplified 82.3%

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

      1. clear-num 82.3%

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

      2. inv-pow 82.3%

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

   7. Applied egg-rr 82.3%

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

      1. unpow-1 82.3%

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

      2. associate-/l/ 80.4%

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

      3. *-commutative 80.4%

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

   9. Simplified 80.4%

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

       1. clear-num 80.9%

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

       2. unpow2 80.9%

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

       3. times-frac 91.2%

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

   11. Applied egg-rr 91.2%

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

   if -4.00000000000000032e91 < c < -5.0000000000000001e-100 or 1.1199999999999999e-142 < c < 1.45e81

   1. Initial program 86.4%

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

   if -5.0000000000000001e-100 < c < 1.1199999999999999e-142

   1. Initial program 59.3%

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

   3. Taylor expanded in c around 0 76.6%

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

      1. +-commutative 76.6%

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

      2. mul-1-neg 76.6%

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

      3. unsub-neg 76.6%

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

      4. associate-/l* 76.8%

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

      5. associate-/r/ 71.0%

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

   5. Simplified 71.0%

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

      1. *-un-lft-identity 71.0%

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

      2. unpow2 71.0%

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

      3. times-frac 74.5%

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

   7. Applied egg-rr 74.5%

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

      1. associate-*l* 81.6%

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

      2. fma-neg 81.6%

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

   9. Applied egg-rr 81.6%

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

       1. fma-udef 81.6%

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

       2. *-lft-identity 81.6%

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

       3. associate-*l/ 81.3%

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

       4. distribute-rgt-neg-in 81.3%

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

       5. distribute-lft-out 81.3%

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

       6. *-commutative 81.3%

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

   11. Simplified 81.3%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{d}{c} \cdot \frac{a}{c}, \frac{b}{c}\right)\\ \mathbf{elif}\;c \leq -5 \cdot 10^{-100}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.12 \cdot 10^{-142}:\\ \;\;\;\;\frac{1}{d} \cdot \left(c \cdot \frac{b}{d} - a\right)\\ \mathbf{elif}\;c \leq 1.45 \cdot 10^{+81}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{d}{c} \cdot \frac{a}{c}, \frac{b}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{c} - \frac{d}{\frac{{c}^{2}}{a}}\\ \mathbf{if}\;c \leq -5.5 \cdot 10^{+88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -6 \cdot 10^{-100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{1}{d} \cdot \left(c \cdot \frac{b}{d} - a\right)\\ \mathbf{elif}\;c \leq 3.5 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d))))
        (t_1 (- (/ b c) (/ d (/ (pow c 2.0) a)))))
   (if (<= c -5.5e+88)
     t_1
     (if (<= c -6e-100)
       t_0
       (if (<= c 7.2e-146)
         (* (/ 1.0 d) (- (* c (/ b d)) a))
         (if (<= c 3.5e+80) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double t_1 = (b / c) - (d / (pow(c, 2.0) / a));
	double tmp;
	if (c <= -5.5e+88) {
		tmp = t_1;
	} else if (c <= -6e-100) {
		tmp = t_0;
	} else if (c <= 7.2e-146) {
		tmp = (1.0 / d) * ((c * (b / d)) - a);
	} else if (c <= 3.5e+80) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
    t_1 = (b / c) - (d / ((c ** 2.0d0) / a))
    if (c <= (-5.5d+88)) then
        tmp = t_1
    else if (c <= (-6d-100)) then
        tmp = t_0
    else if (c <= 7.2d-146) then
        tmp = (1.0d0 / d) * ((c * (b / d)) - a)
    else if (c <= 3.5d+80) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double t_1 = (b / c) - (d / (Math.pow(c, 2.0) / a));
	double tmp;
	if (c <= -5.5e+88) {
		tmp = t_1;
	} else if (c <= -6e-100) {
		tmp = t_0;
	} else if (c <= 7.2e-146) {
		tmp = (1.0 / d) * ((c * (b / d)) - a);
	} else if (c <= 3.5e+80) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
	t_1 = (b / c) - (d / (math.pow(c, 2.0) / a))
	tmp = 0
	if c <= -5.5e+88:
		tmp = t_1
	elif c <= -6e-100:
		tmp = t_0
	elif c <= 7.2e-146:
		tmp = (1.0 / d) * ((c * (b / d)) - a)
	elif c <= 3.5e+80:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b / c) - Float64(d / Float64((c ^ 2.0) / a)))
	tmp = 0.0
	if (c <= -5.5e+88)
		tmp = t_1;
	elseif (c <= -6e-100)
		tmp = t_0;
	elseif (c <= 7.2e-146)
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(c * Float64(b / d)) - a));
	elseif (c <= 3.5e+80)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	t_1 = (b / c) - (d / ((c ^ 2.0) / a));
	tmp = 0.0;
	if (c <= -5.5e+88)
		tmp = t_1;
	elseif (c <= -6e-100)
		tmp = t_0;
	elseif (c <= 7.2e-146)
		tmp = (1.0 / d) * ((c * (b / d)) - a);
	elseif (c <= 3.5e+80)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b / c), $MachinePrecision] - N[(d / N[(N[Power[c, 2.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -5.5e+88], t$95$1, If[LessEqual[c, -6e-100], t$95$0, If[LessEqual[c, 7.2e-146], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.5e+80], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -5.5e88 or 3.49999999999999994e80 < c

   1. Initial program 40.0%

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

   3. Taylor expanded in c around inf 80.9%

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

      1. +-commutative 80.9%

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

      2. mul-1-neg 80.9%

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

      3. unsub-neg 80.9%

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

      4. *-commutative 80.9%

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

      5. associate-/l* 83.9%

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

   5. Simplified 83.9%

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

   if -5.5e88 < c < -6.0000000000000001e-100 or 7.19999999999999957e-146 < c < 3.49999999999999994e80

   1. Initial program 86.4%

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

   if -6.0000000000000001e-100 < c < 7.19999999999999957e-146

   1. Initial program 59.3%

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

   3. Taylor expanded in c around 0 76.6%

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

      1. +-commutative 76.6%

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

      2. mul-1-neg 76.6%

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

      3. unsub-neg 76.6%

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

      4. associate-/l* 76.8%

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

      5. associate-/r/ 71.0%

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

   5. Simplified 71.0%

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

      1. *-un-lft-identity 71.0%

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

      2. unpow2 71.0%

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

      3. times-frac 74.5%

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

   7. Applied egg-rr 74.5%

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

      1. associate-*l* 81.6%

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

      2. fma-neg 81.6%

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

   9. Applied egg-rr 81.6%

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

       1. fma-udef 81.6%

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

       2. *-lft-identity 81.6%

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

       3. associate-*l/ 81.3%

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

       4. distribute-rgt-neg-in 81.3%

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

       5. distribute-lft-out 81.3%

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

       6. *-commutative 81.3%

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

   11. Simplified 81.3%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{\frac{{c}^{2}}{a}}\\ \mathbf{elif}\;c \leq -6 \cdot 10^{-100}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{1}{d} \cdot \left(c \cdot \frac{b}{d} - a\right)\\ \mathbf{elif}\;c \leq 3.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{\frac{{c}^{2}}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1 \cdot 10^{+93}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -7.5 \cdot 10^{-100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{1}{d} \cdot \left(c \cdot \frac{b}{d} - a\right)\\ \mathbf{elif}\;c \leq 3 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (<= c -1e+93)
     (/ b c)
     (if (<= c -7.5e-100)
       t_0
       (if (<= c 7.5e-143)
         (* (/ 1.0 d) (- (* c (/ b d)) a))
         (if (<= c 3e+81) t_0 (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1e+93) {
		tmp = b / c;
	} else if (c <= -7.5e-100) {
		tmp = t_0;
	} else if (c <= 7.5e-143) {
		tmp = (1.0 / d) * ((c * (b / d)) - a);
	} else if (c <= 3e+81) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
    if (c <= (-1d+93)) then
        tmp = b / c
    else if (c <= (-7.5d-100)) then
        tmp = t_0
    else if (c <= 7.5d-143) then
        tmp = (1.0d0 / d) * ((c * (b / d)) - a)
    else if (c <= 3d+81) then
        tmp = t_0
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1e+93) {
		tmp = b / c;
	} else if (c <= -7.5e-100) {
		tmp = t_0;
	} else if (c <= 7.5e-143) {
		tmp = (1.0 / d) * ((c * (b / d)) - a);
	} else if (c <= 3e+81) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -1e+93:
		tmp = b / c
	elif c <= -7.5e-100:
		tmp = t_0
	elif c <= 7.5e-143:
		tmp = (1.0 / d) * ((c * (b / d)) - a)
	elif c <= 3e+81:
		tmp = t_0
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -1e+93)
		tmp = Float64(b / c);
	elseif (c <= -7.5e-100)
		tmp = t_0;
	elseif (c <= 7.5e-143)
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(c * Float64(b / d)) - a));
	elseif (c <= 3e+81)
		tmp = t_0;
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -1e+93)
		tmp = b / c;
	elseif (c <= -7.5e-100)
		tmp = t_0;
	elseif (c <= 7.5e-143)
		tmp = (1.0 / d) * ((c * (b / d)) - a);
	elseif (c <= 3e+81)
		tmp = t_0;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1e+93], N[(b / c), $MachinePrecision], If[LessEqual[c, -7.5e-100], t$95$0, If[LessEqual[c, 7.5e-143], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3e+81], t$95$0, N[(b / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.00000000000000004e93 or 2.99999999999999997e81 < c

   1. Initial program 40.0%

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

   3. Taylor expanded in c around inf 82.3%

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

   if -1.00000000000000004e93 < c < -7.50000000000000015e-100 or 7.5000000000000003e-143 < c < 2.99999999999999997e81

   1. Initial program 86.4%

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

   if -7.50000000000000015e-100 < c < 7.5000000000000003e-143

   1. Initial program 59.3%

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

   3. Taylor expanded in c around 0 76.6%

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

      1. +-commutative 76.6%

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

      2. mul-1-neg 76.6%

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

      3. unsub-neg 76.6%

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

      4. associate-/l* 76.8%

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

      5. associate-/r/ 71.0%

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

   5. Simplified 71.0%

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

      1. *-un-lft-identity 71.0%

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

      2. unpow2 71.0%

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

      3. times-frac 74.5%

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

   7. Applied egg-rr 74.5%

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

      1. associate-*l* 81.6%

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

      2. fma-neg 81.6%

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

   9. Applied egg-rr 81.6%

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

       1. fma-udef 81.6%

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

       2. *-lft-identity 81.6%

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

       3. associate-*l/ 81.3%

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

       4. distribute-rgt-neg-in 81.3%

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

       5. distribute-lft-out 81.3%

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

       6. *-commutative 81.3%

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

   11. Simplified 81.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1 \cdot 10^{+93}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -7.5 \cdot 10^{-100}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{1}{d} \cdot \left(c \cdot \frac{b}{d} - a\right)\\ \mathbf{elif}\;c \leq 3 \cdot 10^{+81}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -8.2 \cdot 10^{+86}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -6 \cdot 10^{-79}:\\ \;\;\;\;\frac{c \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{1}{d} \cdot \left(c \cdot \frac{b}{d} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -8.2e+86)
   (/ b c)
   (if (<= c -6e-79)
     (/ (* c b) (+ (* c c) (* d d)))
     (if (<= c 4.5e+80) (* (/ 1.0 d) (- (* c (/ b d)) a)) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -8.2e+86) {
		tmp = b / c;
	} else if (c <= -6e-79) {
		tmp = (c * b) / ((c * c) + (d * d));
	} else if (c <= 4.5e+80) {
		tmp = (1.0 / d) * ((c * (b / d)) - a);
	} 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 <= (-8.2d+86)) then
        tmp = b / c
    else if (c <= (-6d-79)) then
        tmp = (c * b) / ((c * c) + (d * d))
    else if (c <= 4.5d+80) then
        tmp = (1.0d0 / d) * ((c * (b / d)) - a)
    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 <= -8.2e+86) {
		tmp = b / c;
	} else if (c <= -6e-79) {
		tmp = (c * b) / ((c * c) + (d * d));
	} else if (c <= 4.5e+80) {
		tmp = (1.0 / d) * ((c * (b / d)) - a);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -8.2e+86:
		tmp = b / c
	elif c <= -6e-79:
		tmp = (c * b) / ((c * c) + (d * d))
	elif c <= 4.5e+80:
		tmp = (1.0 / d) * ((c * (b / d)) - a)
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -8.2e+86)
		tmp = Float64(b / c);
	elseif (c <= -6e-79)
		tmp = Float64(Float64(c * b) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 4.5e+80)
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(c * Float64(b / d)) - a));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -8.2e+86)
		tmp = b / c;
	elseif (c <= -6e-79)
		tmp = (c * b) / ((c * c) + (d * d));
	elseif (c <= 4.5e+80)
		tmp = (1.0 / d) * ((c * (b / d)) - a);
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -8.2e+86], N[(b / c), $MachinePrecision], If[LessEqual[c, -6e-79], N[(N[(c * b), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4.5e+80], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -8.1999999999999998e86 or 4.50000000000000007e80 < c

   1. Initial program 40.0%

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

   3. Taylor expanded in c around inf 82.3%

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

   if -8.1999999999999998e86 < c < -5.99999999999999999e-79

   1. Initial program 92.9%

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

   3. Taylor expanded in b around inf 73.2%

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

      1. *-commutative 73.2%

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

   5. Simplified 73.2%

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

   if -5.99999999999999999e-79 < c < 4.50000000000000007e80

   1. Initial program 67.5%

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

   3. Taylor expanded in c around 0 72.6%

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

      1. +-commutative 72.6%

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

      2. mul-1-neg 72.6%

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

      3. unsub-neg 72.6%

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

      4. associate-/l* 72.7%

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

      5. associate-/r/ 69.0%

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

   5. Simplified 69.0%

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

      1. *-un-lft-identity 69.0%

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

      2. unpow2 69.0%

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

      3. times-frac 72.6%

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

   7. Applied egg-rr 72.6%

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

      1. associate-*l* 77.1%

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

      2. fma-neg 77.1%

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

   9. Applied egg-rr 77.1%

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

       1. fma-udef 77.1%

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

       2. *-lft-identity 77.1%

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

       3. associate-*l/ 76.9%

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

       4. distribute-rgt-neg-in 76.9%

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

       5. distribute-lft-out 76.9%

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

       6. *-commutative 76.9%

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

   11. Simplified 76.9%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8.2 \cdot 10^{+86}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -6 \cdot 10^{-79}:\\ \;\;\;\;\frac{c \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{1}{d} \cdot \left(c \cdot \frac{b}{d} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.9 \cdot 10^{-21} \lor \neg \left(d \leq 1.45 \cdot 10^{-34}\right):\\ \;\;\;\;c \cdot \frac{\frac{b}{d}}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -5.9e-21) (not (<= d 1.45e-34)))
   (- (* c (/ (/ b d) d)) (/ a d))
   (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.9e-21) || !(d <= 1.45e-34)) {
		tmp = (c * ((b / d) / d)) - (a / d);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-5.9d-21)) .or. (.not. (d <= 1.45d-34))) then
        tmp = (c * ((b / d) / d)) - (a / d)
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.9e-21) || !(d <= 1.45e-34)) {
		tmp = (c * ((b / d) / d)) - (a / d);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -5.9e-21) or not (d <= 1.45e-34):
		tmp = (c * ((b / d) / d)) - (a / d)
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -5.9e-21) || !(d <= 1.45e-34))
		tmp = Float64(Float64(c * Float64(Float64(b / d) / d)) - Float64(a / d));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -5.9e-21) || ~((d <= 1.45e-34)))
		tmp = (c * ((b / d) / d)) - (a / d);
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -5.9e-21], N[Not[LessEqual[d, 1.45e-34]], $MachinePrecision]], N[(N[(c * N[(N[(b / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -5.9000000000000003e-21 or 1.4500000000000001e-34 < d

   1. Initial program 57.4%

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

   3. Taylor expanded in c around 0 73.7%

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

      1. +-commutative 73.7%

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

      2. mul-1-neg 73.7%

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

      3. unsub-neg 73.7%

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

      4. associate-/l* 75.3%

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

      5. associate-/r/ 76.0%

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

   5. Simplified 76.0%

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

      1. *-un-lft-identity 76.0%

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

      2. unpow2 76.0%

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

      3. times-frac 78.7%

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

   7. Applied egg-rr 78.7%

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

      1. associate-*l/ 78.7%

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

      2. *-lft-identity 78.7%

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

   9. Simplified 78.7%

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

   if -5.9000000000000003e-21 < d < 1.4500000000000001e-34

   1. Initial program 70.7%

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

   3. Taylor expanded in c around inf 68.9%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.9 \cdot 10^{-21} \lor \neg \left(d \leq 1.45 \cdot 10^{-34}\right):\\ \;\;\;\;c \cdot \frac{\frac{b}{d}}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{+45} \lor \neg \left(d \leq 3.9 \cdot 10^{+58}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -5.2e+45) (not (<= d 3.9e+58))) (/ (- a) d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.2e+45) || !(d <= 3.9e+58)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-5.2d+45)) .or. (.not. (d <= 3.9d+58))) then
        tmp = -a / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.2e+45) || !(d <= 3.9e+58)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -5.2e+45) or not (d <= 3.9e+58):
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -5.2e+45) || !(d <= 3.9e+58))
		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 ((d <= -5.2e+45) || ~((d <= 3.9e+58)))
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -5.2e+45], N[Not[LessEqual[d, 3.9e+58]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -5.20000000000000014e45 or 3.9000000000000001e58 < d

   1. Initial program 50.0%

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

   3. Taylor expanded in c around 0 77.0%

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

      1. associate-*r/ 77.0%

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

      2. neg-mul-1 77.0%

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

   5. Simplified 77.0%

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

   if -5.20000000000000014e45 < d < 3.9000000000000001e58

   1. Initial program 72.7%

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

   3. Taylor expanded in c around inf 61.4%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{+45} \lor \neg \left(d \leq 3.9 \cdot 10^{+58}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.5%

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

3. Taylor expanded in c around inf 43.6%

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

4. Final simplification 43.6%

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

Developer target: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024019 
(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))))