Complex division, imag part

Percentage Accurate: 62.2% → 96.0%

Time: 12.1s

Alternatives: 15

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: 62.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.0% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

   1. div-sub 59.2%

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

   2. sub-neg 59.2%

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

   3. *-commutative 59.2%

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

   4. add-sqr-sqrt 59.2%

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

   5. times-frac 60.3%

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

   6. fma-def 60.3%

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

   7. hypot-def 60.3%

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

   8. hypot-def 74.6%

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

   9. associate-/l* 78.2%

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

   10. add-sqr-sqrt 78.2%

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

   11. pow2 78.2%

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

   12. hypot-def 78.2%

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

4. Applied egg-rr 78.2%

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

5. Taylor expanded in c around 0 91.4%

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

   1. +-commutative 91.4%

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

   2. unpow2 91.4%

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

   3. associate-*r/ 97.2%

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

   4. fma-def 97.2%

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

7. Simplified 97.2%

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

8. Final simplification 97.2%

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

Alternative 2: 87.2% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c b) (* d a))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) INFINITY)
     (* (/ 1.0 (hypot c d)) (/ t_0 (hypot c d)))
     (fma 1.0 (/ b (hypot c d)) (/ (/ (- 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))) <= ((double) INFINITY)) {
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	} else {
		tmp = fma(1.0, (b / hypot(c, d)), ((-d / (hypot(c, d) / a)) / 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))) <= Inf)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, d)));
	else
		tmp = fma(1.0, Float64(b / hypot(c, d)), Float64(Float64(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[(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], Infinity], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[(N[((-d) / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 79.8%

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

      1. *-un-lft-identity 79.8%

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

      2. add-sqr-sqrt 79.8%

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

      3. times-frac 79.9%

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

      4. hypot-def 79.9%

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

      5. hypot-def 94.4%

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

   4. Applied egg-rr 94.4%

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

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

   1. Initial program 0.0%

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

      1. div-sub 0.0%

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

      2. sub-neg 0.0%

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

      3. *-commutative 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. times-frac 1.6%

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

      6. fma-def 1.6%

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

      7. hypot-def 1.6%

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

      8. hypot-def 41.4%

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

      9. associate-/l* 46.6%

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

      10. add-sqr-sqrt 46.6%

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

      11. pow2 46.6%

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

      12. hypot-def 46.6%

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

   4. Applied egg-rr 46.6%

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

   6. Applied egg-rr 43.1%

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

      1. associate-*l/ 43.1%

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

      2. *-lft-identity 43.1%

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

      3. associate-/l* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in c around inf 57.6%

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

4. Final simplification 86.1%

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

Alternative 3: 75.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{\mathsf{fma}\left(c, \frac{c}{d}, d\right)}\\ t_1 := \frac{c}{\frac{{d}^{2}}{b}} - \frac{a}{d}\\ \mathbf{if}\;d \leq -5.7 \cdot 10^{+242}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{-c}{d}}{\frac{\mathsf{hypot}\left(c, d\right)}{b}}\\ \mathbf{elif}\;d \leq -58000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-33}:\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{a}{{c}^{2}}\\ \mathbf{elif}\;d \leq -2.4 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-111}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right) \cdot \frac{\mathsf{hypot}\left(c, d\right)}{c}}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{a}{c \cdot \frac{c}{d}}, \frac{b}{c}\right)\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) (fma c (/ c d) d)))
        (t_1 (- (/ c (/ (pow d 2.0) b)) (/ a d))))
   (if (<= d -5.7e+242)
     t_0
     (if (<= d -4.8e+192)
       (/ (/ (- c) d) (/ (hypot c d) b))
       (if (<= d -58000000000.0)
         t_1
         (if (<= d -4e-33)
           (- (/ b c) (* d (/ a (pow c 2.0))))
           (if (<= d -2.4e-69)
             t_0
             (if (<= d -4e-111)
               (/ b (* (hypot c d) (/ (hypot c d) c)))
               (if (<= d 4.4e-164)
                 (fma -1.0 (/ a (* c (/ c d))) (/ b c))
                 (if (<= d 5.5e+45)
                   (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))
                   t_1))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / fma(c, (c / d), d);
	double t_1 = (c / (pow(d, 2.0) / b)) - (a / d);
	double tmp;
	if (d <= -5.7e+242) {
		tmp = t_0;
	} else if (d <= -4.8e+192) {
		tmp = (-c / d) / (hypot(c, d) / b);
	} else if (d <= -58000000000.0) {
		tmp = t_1;
	} else if (d <= -4e-33) {
		tmp = (b / c) - (d * (a / pow(c, 2.0)));
	} else if (d <= -2.4e-69) {
		tmp = t_0;
	} else if (d <= -4e-111) {
		tmp = b / (hypot(c, d) * (hypot(c, d) / c));
	} else if (d <= 4.4e-164) {
		tmp = fma(-1.0, (a / (c * (c / d))), (b / c));
	} else if (d <= 5.5e+45) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / fma(c, Float64(c / d), d))
	t_1 = Float64(Float64(c / Float64((d ^ 2.0) / b)) - Float64(a / d))
	tmp = 0.0
	if (d <= -5.7e+242)
		tmp = t_0;
	elseif (d <= -4.8e+192)
		tmp = Float64(Float64(Float64(-c) / d) / Float64(hypot(c, d) / b));
	elseif (d <= -58000000000.0)
		tmp = t_1;
	elseif (d <= -4e-33)
		tmp = Float64(Float64(b / c) - Float64(d * Float64(a / (c ^ 2.0))));
	elseif (d <= -2.4e-69)
		tmp = t_0;
	elseif (d <= -4e-111)
		tmp = Float64(b / Float64(hypot(c, d) * Float64(hypot(c, d) / c)));
	elseif (d <= 4.4e-164)
		tmp = fma(-1.0, Float64(a / Float64(c * Float64(c / d))), Float64(b / c));
	elseif (d <= 5.5e+45)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / N[(c * N[(c / d), $MachinePrecision] + d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c / N[(N[Power[d, 2.0], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.7e+242], t$95$0, If[LessEqual[d, -4.8e+192], N[(N[((-c) / d), $MachinePrecision] / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -58000000000.0], t$95$1, If[LessEqual[d, -4e-33], N[(N[(b / c), $MachinePrecision] - N[(d * N[(a / N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.4e-69], t$95$0, If[LessEqual[d, -4e-111], N[(b / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] * N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.4e-164], N[(-1.0 * N[(a / N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.5e+45], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if d < -5.7000000000000003e242 or -4.0000000000000002e-33 < d < -2.4000000000000001e-69

   1. Initial program 55.7%

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

      1. div-sub 55.7%

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

      2. sub-neg 55.7%

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

      3. *-commutative 55.7%

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

      4. add-sqr-sqrt 55.7%

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

      5. times-frac 55.9%

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

      6. fma-def 55.9%

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

      7. hypot-def 55.9%

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

      8. hypot-def 65.2%

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

      9. associate-/l* 66.8%

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

      10. add-sqr-sqrt 66.8%

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

      11. pow2 66.8%

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

      12. hypot-def 66.8%

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

   4. Applied egg-rr 66.8%

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

   5. Taylor expanded in c around 0 90.7%

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

      1. +-commutative 90.7%

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

      2. unpow2 90.7%

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

      3. associate-*r/ 99.9%

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

      4. fma-def 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in b around 0 81.4%

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

      1. associate-*r/ 81.4%

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

      2. neg-mul-1 81.4%

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

      3. +-commutative 81.4%

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

      4. unpow2 81.4%

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

      5. associate-*r/ 90.6%

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

      6. fma-udef 90.6%

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

   10. Simplified 90.6%

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

   if -5.7000000000000003e242 < d < -4.7999999999999996e192

   1. Initial program 36.7%

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

   3. Taylor expanded in b around inf 36.7%

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

      1. associate-/l* 38.9%

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

      2. associate-/r/ 38.9%

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

      3. +-commutative 38.9%

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

      4. unpow2 38.9%

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

      5. fma-def 38.9%

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

   5. Simplified 38.9%

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

      1. *-commutative 38.9%

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

      2. associate-*r/ 36.7%

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

      3. fma-udef 36.7%

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

      4. add-sqr-sqrt 36.7%

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

      5. +-commutative 36.7%

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

      6. pow2 36.7%

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

      7. hypot-udef 36.7%

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

      8. +-commutative 36.7%

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

      9. pow2 36.7%

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

      10. hypot-udef 36.7%

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

      11. frac-times 100.0%

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

      12. clear-num 99.9%

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

      13. un-div-inv 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in d around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -4.7999999999999996e192 < d < -5.8e10 or 5.5000000000000001e45 < d

   1. Initial program 45.7%

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

   3. Taylor expanded in c around 0 78.5%

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

      1. +-commutative 78.5%

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

      2. mul-1-neg 78.5%

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

      3. unsub-neg 78.5%

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

      4. *-commutative 78.5%

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

      5. associate-/l* 83.7%

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

   5. Simplified 83.7%

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

   if -5.8e10 < d < -4.0000000000000002e-33

   1. Initial program 46.3%

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

   3. Taylor expanded in c around inf 73.9%

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

      1. +-commutative 73.9%

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

      2. mul-1-neg 73.9%

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

      3. unsub-neg 73.9%

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

      4. associate-/l* 73.8%

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

      5. associate-/r/ 73.8%

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

   5. Simplified 73.8%

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

   if -2.4000000000000001e-69 < d < -4.00000000000000035e-111

   1. Initial program 73.1%

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

   3. Taylor expanded in b around inf 73.1%

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

      1. associate-/l* 86.7%

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

      2. associate-/r/ 73.2%

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

      3. +-commutative 73.2%

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

      4. unpow2 73.2%

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

      5. fma-def 73.2%

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

   5. Simplified 73.2%

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

      1. *-commutative 73.2%

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

      2. associate-*r/ 73.1%

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

      3. fma-udef 73.1%

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

      4. add-sqr-sqrt 73.1%

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

      5. +-commutative 73.1%

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

      6. pow2 73.1%

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

      7. hypot-udef 73.1%

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

      8. +-commutative 73.1%

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

      9. pow2 73.1%

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

      10. hypot-udef 73.1%

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

      11. frac-times 86.4%

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

      12. clear-num 86.4%

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

      13. frac-times 99.8%

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

      14. *-un-lft-identity 99.8%

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

   7. Applied egg-rr 99.8%

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

   if -4.00000000000000035e-111 < d < 4.39999999999999975e-164

   1. Initial program 76.7%

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

   3. Taylor expanded in c around inf 80.3%

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

      1. fma-def 80.3%

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

      2. associate-/l* 79.5%

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

   5. Simplified 79.5%

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

      1. div-inv 79.4%

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

      2. pow2 79.4%

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

      3. associate-*l* 89.5%

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

   7. Applied egg-rr 89.5%

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

   8. Taylor expanded in c around 0 89.5%

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

   if 4.39999999999999975e-164 < d < 5.5000000000000001e45

   1. Initial program 90.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
3. Recombined 7 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.7 \cdot 10^{+242}:\\ \;\;\;\;\frac{-a}{\mathsf{fma}\left(c, \frac{c}{d}, d\right)}\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{-c}{d}}{\frac{\mathsf{hypot}\left(c, d\right)}{b}}\\ \mathbf{elif}\;d \leq -58000000000:\\ \;\;\;\;\frac{c}{\frac{{d}^{2}}{b}} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-33}:\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{a}{{c}^{2}}\\ \mathbf{elif}\;d \leq -2.4 \cdot 10^{-69}:\\ \;\;\;\;\frac{-a}{\mathsf{fma}\left(c, \frac{c}{d}, d\right)}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-111}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right) \cdot \frac{\mathsf{hypot}\left(c, d\right)}{c}}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{a}{c \cdot \frac{c}{d}}, \frac{b}{c}\right)\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{{d}^{2}}{b}} - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.0% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c b) (* d a))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) INFINITY)
     (* (/ 1.0 (hypot c d)) (/ t_0 (hypot c d)))
     (/ (- a) (fma c (/ c d) 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))) <= ((double) INFINITY)) {
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	} else {
		tmp = -a / fma(c, (c / d), 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))) <= Inf)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, d)));
	else
		tmp = Float64(Float64(-a) / fma(c, Float64(c / d), 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], Infinity], 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[((-a) / N[(c * N[(c / d), $MachinePrecision] + 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))) < +inf.0

   1. Initial program 79.8%

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

      1. *-un-lft-identity 79.8%

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

      2. add-sqr-sqrt 79.8%

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

      3. times-frac 79.9%

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

      4. hypot-def 79.9%

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

      5. hypot-def 94.4%

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

   4. Applied egg-rr 94.4%

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

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

   1. Initial program 0.0%

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

      1. div-sub 0.0%

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

      2. sub-neg 0.0%

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

      3. *-commutative 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. times-frac 1.6%

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

      6. fma-def 1.6%

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

      7. hypot-def 1.6%

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

      8. hypot-def 41.4%

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

      9. associate-/l* 46.6%

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

      10. add-sqr-sqrt 46.6%

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

      11. pow2 46.6%

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

      12. hypot-def 46.6%

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

   4. Applied egg-rr 46.6%

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

   5. Taylor expanded in c around 0 87.7%

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

      1. +-commutative 87.7%

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

      2. unpow2 87.7%

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

      3. associate-*r/ 99.9%

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

      4. fma-def 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in b around 0 44.8%

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

      1. associate-*r/ 44.8%

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

      2. neg-mul-1 44.8%

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

      3. +-commutative 44.8%

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

      4. unpow2 44.8%

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

      5. associate-*r/ 57.0%

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

      6. fma-udef 57.0%

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

   10. Simplified 57.0%

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

4. Final simplification 85.9%

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

Alternative 5: 73.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{+34} \lor \neg \left(a \leq 8.5 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{-a}{\mathsf{fma}\left(c, \frac{c}{d}, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{\mathsf{hypot}\left(c, d\right)}}{\frac{\mathsf{hypot}\left(c, d\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= a -3.3e+34) (not (<= a 8.5e-27)))
   (/ (- a) (fma c (/ c d) d))
   (/ (/ b (hypot c d)) (/ (hypot c d) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((a <= -3.3e+34) || !(a <= 8.5e-27)) {
		tmp = -a / fma(c, (c / d), d);
	} else {
		tmp = (b / hypot(c, d)) / (hypot(c, d) / c);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((a <= -3.3e+34) || !(a <= 8.5e-27))
		tmp = Float64(Float64(-a) / fma(c, Float64(c / d), d));
	else
		tmp = Float64(Float64(b / hypot(c, d)) / Float64(hypot(c, d) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[a, -3.3e+34], N[Not[LessEqual[a, 8.5e-27]], $MachinePrecision]], N[((-a) / N[(c * N[(c / d), $MachinePrecision] + d), $MachinePrecision]), $MachinePrecision], N[(N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.29999999999999988e34 or 8.50000000000000033e-27 < a

   1. Initial program 53.2%

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

      1. div-sub 49.2%

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

      2. sub-neg 49.2%

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

      3. *-commutative 49.2%

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

      4. add-sqr-sqrt 49.2%

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

      5. times-frac 50.2%

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

      6. fma-def 50.2%

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

      7. hypot-def 50.2%

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

      8. hypot-def 57.0%

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

      9. associate-/l* 63.9%

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

      10. add-sqr-sqrt 63.9%

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

      11. pow2 63.9%

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

      12. hypot-def 63.9%

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

   4. Applied egg-rr 63.9%

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

   5. Taylor expanded in c around 0 86.3%

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

      1. +-commutative 86.3%

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

      2. unpow2 86.3%

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

      3. associate-*r/ 97.0%

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

      4. fma-def 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in b around 0 64.6%

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

      1. associate-*r/ 64.6%

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

      2. neg-mul-1 64.6%

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

      3. +-commutative 64.6%

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

      4. unpow2 64.6%

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

      5. associate-*r/ 75.1%

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

      6. fma-udef 75.1%

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

   10. Simplified 75.1%

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

   if -3.29999999999999988e34 < a < 8.50000000000000033e-27

   1. Initial program 70.3%

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

   3. Taylor expanded in b around inf 57.7%

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

      1. associate-/l* 58.5%

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

      2. associate-/r/ 55.4%

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

      3. +-commutative 55.4%

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

      4. unpow2 55.4%

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

      5. fma-def 55.4%

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

   5. Simplified 55.4%

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

      1. *-commutative 55.4%

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

      2. associate-*r/ 57.7%

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

      3. fma-udef 57.7%

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

      4. add-sqr-sqrt 57.7%

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

      5. +-commutative 57.7%

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

      6. pow2 57.7%

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

      7. hypot-udef 57.7%

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

      8. +-commutative 57.7%

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

      9. pow2 57.7%

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

      10. hypot-udef 57.7%

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

      11. frac-times 83.2%

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

      12. clear-num 82.7%

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

      13. associate-*l/ 82.7%

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

      14. *-un-lft-identity 82.7%

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

   7. Applied egg-rr 82.7%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{+34} \lor \neg \left(a \leq 8.5 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{-a}{\mathsf{fma}\left(c, \frac{c}{d}, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{\mathsf{hypot}\left(c, d\right)}}{\frac{\mathsf{hypot}\left(c, d\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{+31} \lor \neg \left(a \leq 4.7 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{-a}{\mathsf{fma}\left(c, \frac{c}{d}, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\mathsf{hypot}\left(c, d\right)}}{\frac{\mathsf{hypot}\left(c, d\right)}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= a -3.5e+31) (not (<= a 4.7e-38)))
   (/ (- a) (fma c (/ c d) d))
   (/ (/ c (hypot c d)) (/ (hypot c d) b))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((a <= -3.5e+31) || !(a <= 4.7e-38)) {
		tmp = -a / fma(c, (c / d), d);
	} else {
		tmp = (c / hypot(c, d)) / (hypot(c, d) / b);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((a <= -3.5e+31) || !(a <= 4.7e-38))
		tmp = Float64(Float64(-a) / fma(c, Float64(c / d), d));
	else
		tmp = Float64(Float64(c / hypot(c, d)) / Float64(hypot(c, d) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[a, -3.5e+31], N[Not[LessEqual[a, 4.7e-38]], $MachinePrecision]], N[((-a) / N[(c * N[(c / d), $MachinePrecision] + d), $MachinePrecision]), $MachinePrecision], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.5e31 or 4.69999999999999998e-38 < a

   1. Initial program 53.9%

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

      1. div-sub 50.0%

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

      2. sub-neg 50.0%

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

      3. *-commutative 50.0%

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

      4. add-sqr-sqrt 50.0%

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

      5. times-frac 51.0%

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

      6. fma-def 51.0%

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

      7. hypot-def 51.0%

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

      8. hypot-def 58.3%

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

      9. associate-/l* 65.0%

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

      10. add-sqr-sqrt 65.0%

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

      11. pow2 65.0%

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

      12. hypot-def 65.0%

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

   4. Applied egg-rr 65.0%

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

   5. Taylor expanded in c around 0 86.7%

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

      1. +-commutative 86.7%

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

      2. unpow2 86.7%

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

      3. associate-*r/ 97.1%

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

      4. fma-def 97.1%

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

   7. Simplified 97.1%

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

   8. Taylor expanded in b around 0 64.2%

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

      1. associate-*r/ 64.2%

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

      2. neg-mul-1 64.2%

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

      3. +-commutative 64.2%

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

      4. unpow2 64.2%

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

      5. associate-*r/ 74.3%

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

      6. fma-udef 74.3%

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

   10. Simplified 74.3%

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

   if -3.5e31 < a < 4.69999999999999998e-38

   1. Initial program 70.2%

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

   3. Taylor expanded in b around inf 58.8%

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

      1. associate-/l* 59.5%

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

      2. associate-/r/ 56.3%

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

      3. +-commutative 56.3%

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

      4. unpow2 56.3%

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

      5. fma-def 56.3%

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

   5. Simplified 56.3%

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

      1. *-commutative 56.3%

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

      2. associate-*r/ 58.8%

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

      3. fma-udef 58.8%

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

      4. add-sqr-sqrt 58.8%

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

      5. +-commutative 58.8%

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

      6. pow2 58.8%

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

      7. hypot-udef 58.8%

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

      8. +-commutative 58.8%

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

      9. pow2 58.8%

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

      10. hypot-udef 58.8%

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

      11. frac-times 84.3%

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

      12. clear-num 83.8%

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

      13. un-div-inv 84.6%

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

   7. Applied egg-rr 84.6%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{+31} \lor \neg \left(a \leq 4.7 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{-a}{\mathsf{fma}\left(c, \frac{c}{d}, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\mathsf{hypot}\left(c, d\right)}}{\frac{\mathsf{hypot}\left(c, d\right)}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-321} \lor \neg \left(t_0 \leq 0\right) \land t_0 \leq 5 \cdot 10^{+263}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\mathsf{fma}\left(c, \frac{c}{d}, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (or (<= t_0 -5e-321) (and (not (<= t_0 0.0)) (<= t_0 5e+263)))
     t_0
     (/ (- a) (fma c (/ c d) 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 ((t_0 <= -5e-321) || (!(t_0 <= 0.0) && (t_0 <= 5e+263))) {
		tmp = t_0;
	} else {
		tmp = -a / fma(c, (c / d), 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 ((t_0 <= -5e-321) || (!(t_0 <= 0.0) && (t_0 <= 5e+263)))
		tmp = t_0;
	else
		tmp = Float64(Float64(-a) / fma(c, Float64(c / d), 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[Or[LessEqual[t$95$0, -5e-321], And[N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision], LessEqual[t$95$0, 5e+263]]], t$95$0, N[((-a) / N[(c * N[(c / d), $MachinePrecision] + 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))) < -4.99994e-321 or -0.0 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 5.00000000000000022e263

   1. Initial program 92.1%

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

   if -4.99994e-321 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < -0.0 or 5.00000000000000022e263 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 34.0%

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

      1. div-sub 30.9%

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

      2. sub-neg 30.9%

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

      3. *-commutative 30.9%

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

      4. add-sqr-sqrt 30.9%

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

      5. times-frac 33.7%

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

      6. fma-def 33.7%

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

      7. hypot-def 33.7%

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

      8. hypot-def 59.5%

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

      9. associate-/l* 64.7%

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

      10. add-sqr-sqrt 64.7%

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

      11. pow2 64.7%

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

      12. hypot-def 64.7%

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

   4. Applied egg-rr 64.7%

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

   5. Taylor expanded in c around 0 88.4%

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

      1. +-commutative 88.4%

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

      2. unpow2 88.4%

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

      3. associate-*r/ 97.9%

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

      4. fma-def 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in b around 0 58.9%

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

      1. associate-*r/ 58.9%

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

      2. neg-mul-1 58.9%

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

      3. +-commutative 58.9%

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

      4. unpow2 58.9%

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

      5. associate-*r/ 68.2%

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

      6. fma-udef 68.2%

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

   10. Simplified 68.2%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq -5 \cdot 10^{-321} \lor \neg \left(\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 0\right) \land \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+263}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\mathsf{fma}\left(c, \frac{c}{d}, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{\frac{{d}^{2}}{b}} - \frac{a}{d}\\ \mathbf{if}\;d \leq -5.7 \cdot 10^{+242}:\\ \;\;\;\;\frac{-a}{\mathsf{fma}\left(c, \frac{c}{d}, d\right)}\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{-c}{d}}{\frac{\mathsf{hypot}\left(c, d\right)}{b}}\\ \mathbf{elif}\;d \leq -1000000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{a}{c \cdot \frac{c}{d}}, \frac{b}{c}\right)\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{+45}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (/ c (/ (pow d 2.0) b)) (/ a d))))
   (if (<= d -5.7e+242)
     (/ (- a) (fma c (/ c d) d))
     (if (<= d -4.8e+192)
       (/ (/ (- c) d) (/ (hypot c d) b))
       (if (<= d -1000000000000.0)
         t_0
         (if (<= d 2.1e-164)
           (fma -1.0 (/ a (* c (/ c d))) (/ b c))
           (if (<= d 5.8e+45)
             (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))
             t_0)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c / (pow(d, 2.0) / b)) - (a / d);
	double tmp;
	if (d <= -5.7e+242) {
		tmp = -a / fma(c, (c / d), d);
	} else if (d <= -4.8e+192) {
		tmp = (-c / d) / (hypot(c, d) / b);
	} else if (d <= -1000000000000.0) {
		tmp = t_0;
	} else if (d <= 2.1e-164) {
		tmp = fma(-1.0, (a / (c * (c / d))), (b / c));
	} else if (d <= 5.8e+45) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c / Float64((d ^ 2.0) / b)) - Float64(a / d))
	tmp = 0.0
	if (d <= -5.7e+242)
		tmp = Float64(Float64(-a) / fma(c, Float64(c / d), d));
	elseif (d <= -4.8e+192)
		tmp = Float64(Float64(Float64(-c) / d) / Float64(hypot(c, d) / b));
	elseif (d <= -1000000000000.0)
		tmp = t_0;
	elseif (d <= 2.1e-164)
		tmp = fma(-1.0, Float64(a / Float64(c * Float64(c / d))), Float64(b / c));
	elseif (d <= 5.8e+45)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c / N[(N[Power[d, 2.0], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.7e+242], N[((-a) / N[(c * N[(c / d), $MachinePrecision] + d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.8e+192], N[(N[((-c) / d), $MachinePrecision] / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1000000000000.0], t$95$0, If[LessEqual[d, 2.1e-164], N[(-1.0 * N[(a / N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.8e+45], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -5.7000000000000003e242

   1. Initial program 37.2%

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

      1. div-sub 37.2%

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

      2. sub-neg 37.2%

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

      3. *-commutative 37.2%

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

      4. add-sqr-sqrt 37.2%

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

      5. times-frac 37.2%

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

      6. fma-def 37.2%

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

      7. hypot-def 37.2%

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

      8. hypot-def 37.2%

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

      9. associate-/l* 39.8%

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

      10. add-sqr-sqrt 39.8%

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

      11. pow2 39.8%

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

      12. hypot-def 39.8%

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

   4. Applied egg-rr 39.8%

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

   5. Taylor expanded in c around 0 83.1%

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

      1. +-commutative 83.1%

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

      2. unpow2 83.1%

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

      3. associate-*r/ 100.0%

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

      4. fma-def 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in b around 0 83.1%

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

      1. associate-*r/ 83.1%

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

      2. neg-mul-1 83.1%

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

      3. +-commutative 83.1%

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

      4. unpow2 83.1%

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

      5. associate-*r/ 100.0%

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

      6. fma-udef 100.0%

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

   10. Simplified 100.0%

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

   if -5.7000000000000003e242 < d < -4.7999999999999996e192

   1. Initial program 36.7%

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

   3. Taylor expanded in b around inf 36.7%

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

      1. associate-/l* 38.9%

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

      2. associate-/r/ 38.9%

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

      3. +-commutative 38.9%

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

      4. unpow2 38.9%

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

      5. fma-def 38.9%

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

   5. Simplified 38.9%

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

      1. *-commutative 38.9%

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

      2. associate-*r/ 36.7%

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

      3. fma-udef 36.7%

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

      4. add-sqr-sqrt 36.7%

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

      5. +-commutative 36.7%

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

      6. pow2 36.7%

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

      7. hypot-udef 36.7%

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

      8. +-commutative 36.7%

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

      9. pow2 36.7%

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

      10. hypot-udef 36.7%

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

      11. frac-times 100.0%

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

      12. clear-num 99.9%

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

      13. un-div-inv 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in d around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -4.7999999999999996e192 < d < -1e12 or 5.7999999999999994e45 < d

   1. Initial program 45.7%

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

   3. Taylor expanded in c around 0 78.5%

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

      1. +-commutative 78.5%

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

      2. mul-1-neg 78.5%

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

      3. unsub-neg 78.5%

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

      4. *-commutative 78.5%

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

      5. associate-/l* 83.7%

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

   5. Simplified 83.7%

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

   if -1e12 < d < 2.0999999999999999e-164

   1. Initial program 73.2%

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

   3. Taylor expanded in c around inf 76.1%

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

      1. fma-def 76.1%

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

      2. associate-/l* 75.5%

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

   5. Simplified 75.5%

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

      1. div-inv 75.5%

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

      2. pow2 75.5%

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

      3. associate-*l* 82.7%

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

   7. Applied egg-rr 82.7%

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

   8. Taylor expanded in c around 0 82.8%

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

   if 2.0999999999999999e-164 < d < 5.7999999999999994e45

   1. Initial program 90.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
3. Recombined 5 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.7 \cdot 10^{+242}:\\ \;\;\;\;\frac{-a}{\mathsf{fma}\left(c, \frac{c}{d}, d\right)}\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{-c}{d}}{\frac{\mathsf{hypot}\left(c, d\right)}{b}}\\ \mathbf{elif}\;d \leq -1000000000000:\\ \;\;\;\;\frac{c}{\frac{{d}^{2}}{b}} - \frac{a}{d}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{a}{c \cdot \frac{c}{d}}, \frac{b}{c}\right)\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{+45}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{{d}^{2}}{b}} - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.7 \cdot 10^{+242}:\\ \;\;\;\;\frac{-a}{\mathsf{fma}\left(c, \frac{c}{d}, d\right)}\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{-c}{d}}{\frac{\mathsf{hypot}\left(c, d\right)}{b}}\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{+63} \lor \neg \left(d \leq 3.8 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{c}{\frac{{d}^{2}}{b}} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -5.7e+242)
   (/ (- a) (fma c (/ c d) d))
   (if (<= d -4.8e+192)
     (/ (/ (- c) d) (/ (hypot c d) b))
     (if (or (<= d -1.15e+63) (not (<= d 3.8e+45)))
       (- (/ c (/ (pow d 2.0) b)) (/ a d))
       (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -5.7e+242) {
		tmp = -a / fma(c, (c / d), d);
	} else if (d <= -4.8e+192) {
		tmp = (-c / d) / (hypot(c, d) / b);
	} else if ((d <= -1.15e+63) || !(d <= 3.8e+45)) {
		tmp = (c / (pow(d, 2.0) / b)) - (a / d);
	} else {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -5.7e+242)
		tmp = Float64(Float64(-a) / fma(c, Float64(c / d), d));
	elseif (d <= -4.8e+192)
		tmp = Float64(Float64(Float64(-c) / d) / Float64(hypot(c, d) / b));
	elseif ((d <= -1.15e+63) || !(d <= 3.8e+45))
		tmp = Float64(Float64(c / Float64((d ^ 2.0) / b)) - Float64(a / d));
	else
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -5.7e+242], N[((-a) / N[(c * N[(c / d), $MachinePrecision] + d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.8e+192], N[(N[((-c) / d), $MachinePrecision] / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[d, -1.15e+63], N[Not[LessEqual[d, 3.8e+45]], $MachinePrecision]], N[(N[(c / N[(N[Power[d, 2.0], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.7000000000000003e242

   1. Initial program 37.2%

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

      1. div-sub 37.2%

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

      2. sub-neg 37.2%

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

      3. *-commutative 37.2%

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

      4. add-sqr-sqrt 37.2%

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

      5. times-frac 37.2%

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

      6. fma-def 37.2%

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

      7. hypot-def 37.2%

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

      8. hypot-def 37.2%

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

      9. associate-/l* 39.8%

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

      10. add-sqr-sqrt 39.8%

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

      11. pow2 39.8%

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

      12. hypot-def 39.8%

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

   4. Applied egg-rr 39.8%

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

   5. Taylor expanded in c around 0 83.1%

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

      1. +-commutative 83.1%

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

      2. unpow2 83.1%

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

      3. associate-*r/ 100.0%

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

      4. fma-def 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in b around 0 83.1%

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

      1. associate-*r/ 83.1%

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

      2. neg-mul-1 83.1%

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

      3. +-commutative 83.1%

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

      4. unpow2 83.1%

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

      5. associate-*r/ 100.0%

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

      6. fma-udef 100.0%

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

   10. Simplified 100.0%

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

   if -5.7000000000000003e242 < d < -4.7999999999999996e192

   1. Initial program 36.7%

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

   3. Taylor expanded in b around inf 36.7%

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

      1. associate-/l* 38.9%

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

      2. associate-/r/ 38.9%

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

      3. +-commutative 38.9%

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

      4. unpow2 38.9%

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

      5. fma-def 38.9%

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

   5. Simplified 38.9%

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

      1. *-commutative 38.9%

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

      2. associate-*r/ 36.7%

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

      3. fma-udef 36.7%

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

      4. add-sqr-sqrt 36.7%

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

      5. +-commutative 36.7%

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

      6. pow2 36.7%

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

      7. hypot-udef 36.7%

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

      8. +-commutative 36.7%

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

      9. pow2 36.7%

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

      10. hypot-udef 36.7%

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

      11. frac-times 100.0%

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

      12. clear-num 99.9%

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

      13. un-div-inv 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in d around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -4.7999999999999996e192 < d < -1.14999999999999997e63 or 3.8000000000000002e45 < d

   1. Initial program 42.8%

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

   3. Taylor expanded in c around 0 78.0%

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

      1. +-commutative 78.0%

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

      2. mul-1-neg 78.0%

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

      3. unsub-neg 78.0%

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

      4. *-commutative 78.0%

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

      5. associate-/l* 83.5%

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

   5. Simplified 83.5%

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

   if -1.14999999999999997e63 < d < 3.8000000000000002e45

   1. Initial program 78.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
3. Recombined 4 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.7 \cdot 10^{+242}:\\ \;\;\;\;\frac{-a}{\mathsf{fma}\left(c, \frac{c}{d}, d\right)}\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{-c}{d}}{\frac{\mathsf{hypot}\left(c, d\right)}{b}}\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{+63} \lor \neg \left(d \leq 3.8 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{c}{\frac{{d}^{2}}{b}} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -190000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.95 \cdot 10^{-105}:\\ \;\;\;\;\frac{c \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -190000000.0)
     t_0
     (if (<= d -1.5e-33)
       (/ b c)
       (if (<= d -1e-68)
         t_0
         (if (<= d -1.95e-105)
           (/ (* c b) (+ (* c c) (* d d)))
           (if (<= d 5.8e-18) (/ b c) t_0)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -190000000.0) {
		tmp = t_0;
	} else if (d <= -1.5e-33) {
		tmp = b / c;
	} else if (d <= -1e-68) {
		tmp = t_0;
	} else if (d <= -1.95e-105) {
		tmp = (c * b) / ((c * c) + (d * d));
	} else if (d <= 5.8e-18) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -a / d
    if (d <= (-190000000.0d0)) then
        tmp = t_0
    else if (d <= (-1.5d-33)) then
        tmp = b / c
    else if (d <= (-1d-68)) then
        tmp = t_0
    else if (d <= (-1.95d-105)) then
        tmp = (c * b) / ((c * c) + (d * d))
    else if (d <= 5.8d-18) then
        tmp = b / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -190000000.0) {
		tmp = t_0;
	} else if (d <= -1.5e-33) {
		tmp = b / c;
	} else if (d <= -1e-68) {
		tmp = t_0;
	} else if (d <= -1.95e-105) {
		tmp = (c * b) / ((c * c) + (d * d));
	} else if (d <= 5.8e-18) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	tmp = 0
	if d <= -190000000.0:
		tmp = t_0
	elif d <= -1.5e-33:
		tmp = b / c
	elif d <= -1e-68:
		tmp = t_0
	elif d <= -1.95e-105:
		tmp = (c * b) / ((c * c) + (d * d))
	elif d <= 5.8e-18:
		tmp = b / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -190000000.0)
		tmp = t_0;
	elseif (d <= -1.5e-33)
		tmp = Float64(b / c);
	elseif (d <= -1e-68)
		tmp = t_0;
	elseif (d <= -1.95e-105)
		tmp = Float64(Float64(c * b) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 5.8e-18)
		tmp = Float64(b / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -a / d;
	tmp = 0.0;
	if (d <= -190000000.0)
		tmp = t_0;
	elseif (d <= -1.5e-33)
		tmp = b / c;
	elseif (d <= -1e-68)
		tmp = t_0;
	elseif (d <= -1.95e-105)
		tmp = (c * b) / ((c * c) + (d * d));
	elseif (d <= 5.8e-18)
		tmp = b / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -190000000.0], t$95$0, If[LessEqual[d, -1.5e-33], N[(b / c), $MachinePrecision], If[LessEqual[d, -1e-68], t$95$0, If[LessEqual[d, -1.95e-105], N[(N[(c * b), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.8e-18], N[(b / c), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.9e8 or -1.5000000000000001e-33 < d < -1.00000000000000007e-68 or 5.8e-18 < d

   1. Initial program 48.8%

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

   3. Taylor expanded in c around 0 69.7%

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

      1. associate-*r/ 69.7%

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

      2. neg-mul-1 69.7%

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

   5. Simplified 69.7%

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

   if -1.9e8 < d < -1.5000000000000001e-33 or -1.95e-105 < d < 5.8e-18

   1. Initial program 76.8%

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

   3. Taylor expanded in c around inf 68.3%

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

   if -1.00000000000000007e-68 < d < -1.95e-105

   1. Initial program 80.7%

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

   3. Taylor expanded in b around inf 80.7%

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

      1. *-commutative 80.7%

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

   5. Simplified 80.7%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -190000000:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-68}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1.95 \cdot 10^{-105}:\\ \;\;\;\;\frac{c \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -76000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -8.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq -3.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{d \cdot \left(-a\right)}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-16}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -76000000.0)
     t_0
     (if (<= d -8.2e-34)
       (/ b c)
       (if (<= d -3.8e-69)
         (/ (* d (- a)) (+ (* c c) (* d d)))
         (if (<= d 2.7e-16) (/ b c) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -76000000.0) {
		tmp = t_0;
	} else if (d <= -8.2e-34) {
		tmp = b / c;
	} else if (d <= -3.8e-69) {
		tmp = (d * -a) / ((c * c) + (d * d));
	} else if (d <= 2.7e-16) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -a / d
    if (d <= (-76000000.0d0)) then
        tmp = t_0
    else if (d <= (-8.2d-34)) then
        tmp = b / c
    else if (d <= (-3.8d-69)) then
        tmp = (d * -a) / ((c * c) + (d * d))
    else if (d <= 2.7d-16) then
        tmp = b / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -76000000.0) {
		tmp = t_0;
	} else if (d <= -8.2e-34) {
		tmp = b / c;
	} else if (d <= -3.8e-69) {
		tmp = (d * -a) / ((c * c) + (d * d));
	} else if (d <= 2.7e-16) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	tmp = 0
	if d <= -76000000.0:
		tmp = t_0
	elif d <= -8.2e-34:
		tmp = b / c
	elif d <= -3.8e-69:
		tmp = (d * -a) / ((c * c) + (d * d))
	elif d <= 2.7e-16:
		tmp = b / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -76000000.0)
		tmp = t_0;
	elseif (d <= -8.2e-34)
		tmp = Float64(b / c);
	elseif (d <= -3.8e-69)
		tmp = Float64(Float64(d * Float64(-a)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 2.7e-16)
		tmp = Float64(b / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -a / d;
	tmp = 0.0;
	if (d <= -76000000.0)
		tmp = t_0;
	elseif (d <= -8.2e-34)
		tmp = b / c;
	elseif (d <= -3.8e-69)
		tmp = (d * -a) / ((c * c) + (d * d));
	elseif (d <= 2.7e-16)
		tmp = b / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -76000000.0], t$95$0, If[LessEqual[d, -8.2e-34], N[(b / c), $MachinePrecision], If[LessEqual[d, -3.8e-69], N[(N[(d * (-a)), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.7e-16], N[(b / c), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.6e7 or 2.69999999999999999e-16 < d

   1. Initial program 46.8%

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

   3. Taylor expanded in c around 0 69.8%

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

      1. associate-*r/ 69.8%

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

      2. neg-mul-1 69.8%

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

   5. Simplified 69.8%

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

   if -7.6e7 < d < -8.2000000000000007e-34 or -3.7999999999999998e-69 < d < 2.69999999999999999e-16

   1. Initial program 77.0%

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

   3. Taylor expanded in c around inf 68.1%

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

   if -8.2000000000000007e-34 < d < -3.7999999999999998e-69

   1. Initial program 78.4%

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

   3. Taylor expanded in b around 0 78.7%

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

      1. associate-*r* 78.7%

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

      2. neg-mul-1 78.7%

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

      3. *-commutative 78.7%

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

   5. Simplified 78.7%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -76000000:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -8.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq -3.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{d \cdot \left(-a\right)}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-16}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.6 \cdot 10^{+121} \lor \neg \left(d \leq 8.2 \cdot 10^{+107}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4.6e+121) (not (<= d 8.2e+107)))
   (/ (- a) d)
   (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.6e+121) || !(d <= 8.2e+107)) {
		tmp = -a / d;
	} else {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * 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 ((d <= (-4.6d+121)) .or. (.not. (d <= 8.2d+107))) then
        tmp = -a / d
    else
        tmp = ((c * b) - (d * a)) / ((c * c) + (d * d))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.6e+121) || !(d <= 8.2e+107)) {
		tmp = -a / d;
	} else {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4.6e+121) or not (d <= 8.2e+107):
		tmp = -a / d
	else:
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4.6e+121) || !(d <= 8.2e+107))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -4.6e+121) || ~((d <= 8.2e+107)))
		tmp = -a / d;
	else
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4.6e+121], N[Not[LessEqual[d, 8.2e+107]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.5999999999999997e121 or 8.1999999999999998e107 < d

   1. Initial program 33.6%

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

   3. Taylor expanded in c around 0 78.8%

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

      1. associate-*r/ 78.8%

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

      2. neg-mul-1 78.8%

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

   5. Simplified 78.8%

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

   if -4.5999999999999997e121 < d < 8.1999999999999998e107

   1. Initial program 77.2%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.6 \cdot 10^{+121} \lor \neg \left(d \leq 8.2 \cdot 10^{+107}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1800000 \lor \neg \left(d \leq 4.1 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1800000.0) (not (<= d 4.1e-19))) (/ (- a) d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1800000.0) || !(d <= 4.1e-19)) {
		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 <= (-1800000.0d0)) .or. (.not. (d <= 4.1d-19))) 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 <= -1800000.0) || !(d <= 4.1e-19)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1800000.0) or not (d <= 4.1e-19):
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1800000.0) || !(d <= 4.1e-19))
		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 <= -1800000.0) || ~((d <= 4.1e-19)))
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1800000.0], N[Not[LessEqual[d, 4.1e-19]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.8e6 or 4.09999999999999985e-19 < d

   1. Initial program 46.8%

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

   3. Taylor expanded in c around 0 69.8%

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

      1. associate-*r/ 69.8%

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

      2. neg-mul-1 69.8%

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

   5. Simplified 69.8%

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

   if -1.8e6 < d < 4.09999999999999985e-19

   1. Initial program 77.1%

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

   3. Taylor expanded in c around inf 65.1%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1800000 \lor \neg \left(d \leq 4.1 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 1.4 \cdot 10^{+143}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (if (<= d 1.4e+143) (/ b c) (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 1.4e+143) {
		tmp = b / c;
	} else {
		tmp = b / 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 (d <= 1.4d+143) then
        tmp = b / c
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 1.4e+143) {
		tmp = b / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= 1.4e+143:
		tmp = b / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= 1.4e+143)
		tmp = Float64(b / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= 1.4e+143)
		tmp = b / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, 1.4e+143], N[(b / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.39999999999999999e143

   1. Initial program 69.0%

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

   3. Taylor expanded in c around inf 45.4%

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

   if 1.39999999999999999e143 < d

   1. Initial program 28.5%

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

   3. Taylor expanded in b around inf 27.7%

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

      1. associate-/l* 28.5%

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

      2. associate-/r/ 28.5%

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

      3. +-commutative 28.5%

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

      4. unpow2 28.5%

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

      5. fma-def 28.5%

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

   5. Simplified 28.5%

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

      1. *-commutative 28.5%

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

      2. associate-*r/ 27.7%

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

      3. fma-udef 27.7%

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

      4. add-sqr-sqrt 27.7%

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

      5. +-commutative 27.7%

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

      6. pow2 27.7%

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

      7. hypot-udef 27.7%

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

      8. +-commutative 27.7%

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

      9. pow2 27.7%

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

      10. hypot-udef 27.7%

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

      11. frac-times 40.9%

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

      12. clear-num 40.9%

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

      13. un-div-inv 40.9%

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

   7. Applied egg-rr 40.9%

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

   8. Taylor expanded in c around inf 20.2%

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

   9. Taylor expanded in c around 0 20.2%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.4 \cdot 10^{+143}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.3% 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 61.7%

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

3. Taylor expanded in c around inf 38.3%

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

4. Final simplification 38.3%

   \[\leadsto \frac{b}{c} \]
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 2024018 
(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))))