Complex division, imag part

Percentage Accurate: 61.2% → 95.3%

Time: 12.8s

Alternatives: 16

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.2% 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: 95.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{+86} \lor \neg \left(d \leq 8.8 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b \cdot \frac{c}{d} - a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-\frac{a \cdot d}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.1e+86) (not (<= d 8.8e+136)))
   (* (/ d (hypot d c)) (/ (- (* b (/ c d)) a) (hypot d c)))
   (fma
    (/ c (hypot c d))
    (/ b (hypot c d))
    (/ (- (/ (* a d) (hypot d c))) (hypot d c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.1e+86) || !(d <= 8.8e+136)) {
		tmp = (d / hypot(d, c)) * (((b * (c / d)) - a) / hypot(d, c));
	} else {
		tmp = fma((c / hypot(c, d)), (b / hypot(c, d)), (-((a * d) / hypot(d, c)) / hypot(d, c)));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.1e+86) || !(d <= 8.8e+136))
		tmp = Float64(Float64(d / hypot(d, c)) * Float64(Float64(Float64(b * Float64(c / d)) - a) / hypot(d, c)));
	else
		tmp = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(Float64(-Float64(Float64(a * d) / hypot(d, c))) / hypot(d, c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.1e+86], N[Not[LessEqual[d, 8.8e+136]], $MachinePrecision]], N[(N[(d / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[((-N[(N[(a * d), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]) / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.10000000000000002e86 or 8.7999999999999998e136 < d

   1. Initial program 35.1%

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

   3. Taylor expanded in d around inf 35.1%

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

      1. associate-/l* 32.9%

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

   5. Simplified 32.9%

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

      1. *-commutative 32.9%

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

      2. +-commutative 32.9%

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

      3. add-sqr-sqrt 32.9%

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

      4. hypot-undefine 32.9%

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

      5. hypot-undefine 32.9%

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

      6. times-frac 97.6%

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

   7. Applied egg-rr 97.6%

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

   if -1.10000000000000002e86 < d < 8.7999999999999998e136

   1. Initial program 74.4%

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

      1. div-sub 72.5%

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

      2. *-commutative 72.5%

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

      3. add-sqr-sqrt 72.5%

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

      4. times-frac 74.1%

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

      5. fma-neg 74.1%

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

      6. hypot-define 74.1%

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

      7. hypot-define 93.3%

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

      8. associate-/l* 91.7%

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

      9. add-sqr-sqrt 91.7%

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

      10. pow2 91.7%

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

      11. hypot-define 91.7%

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

   4. Applied egg-rr 91.7%

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

      1. associate-*r/ 93.3%

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

      2. remove-double-neg 93.3%

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

      3. distribute-lft-neg-out 93.3%

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

      4. unpow2 93.3%

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

      5. associate-/r* 98.1%

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

      6. distribute-lft-neg-out 98.1%

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

      7. remove-double-neg 98.1%

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

      8. *-commutative 98.1%

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

      9. hypot-undefine 93.4%

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

      10. +-commutative 93.4%

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

      11. hypot-define 98.1%

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

      12. hypot-undefine 93.4%

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

      13. +-commutative 93.4%

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

      14. hypot-define 98.1%

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

   6. Applied egg-rr 98.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{+86} \lor \neg \left(d \leq 8.8 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b \cdot \frac{c}{d} - a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-\frac{a \cdot d}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+265}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_1, \frac{-\frac{a \cdot d}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_1, a \cdot \left(\frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{-1}{\mathsf{hypot}\left(d, c\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ c (hypot c d))) (t_1 (/ b (hypot c d))))
   (if (<= (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) 2e+265)
     (fma t_0 t_1 (/ (- (/ (* a d) (hypot d c))) (hypot d c)))
     (fma t_0 t_1 (* a (* (/ d (hypot d c)) (/ -1.0 (hypot d c))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = c / hypot(c, d);
	double t_1 = b / hypot(c, d);
	double tmp;
	if ((((b * c) - (a * d)) / ((c * c) + (d * d))) <= 2e+265) {
		tmp = fma(t_0, t_1, (-((a * d) / hypot(d, c)) / hypot(d, c)));
	} else {
		tmp = fma(t_0, t_1, (a * ((d / hypot(d, c)) * (-1.0 / hypot(d, c)))));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(c / hypot(c, d))
	t_1 = Float64(b / hypot(c, d))
	tmp = 0.0
	if (Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d))) <= 2e+265)
		tmp = fma(t_0, t_1, Float64(Float64(-Float64(Float64(a * d) / hypot(d, c))) / hypot(d, c)));
	else
		tmp = fma(t_0, t_1, Float64(a * Float64(Float64(d / hypot(d, c)) * Float64(-1.0 / hypot(d, c)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+265], N[(t$95$0 * t$95$1 + N[((-N[(N[(a * d), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]) / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$1 + N[(a * N[(N[(d / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 2.00000000000000013e265

   1. Initial program 79.5%

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

      1. div-sub 79.5%

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

      2. *-commutative 79.5%

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

      3. add-sqr-sqrt 79.5%

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

      4. times-frac 79.1%

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

      5. fma-neg 79.1%

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

      6. hypot-define 79.1%

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

      7. hypot-define 88.0%

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

      8. associate-/l* 87.0%

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

      9. add-sqr-sqrt 86.9%

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

      10. pow2 86.9%

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

      11. hypot-define 86.9%

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

   4. Applied egg-rr 86.9%

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

      1. associate-*r/ 88.0%

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

      2. remove-double-neg 88.0%

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

      3. distribute-lft-neg-out 88.0%

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

      4. unpow2 88.0%

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

      5. associate-/r* 97.3%

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

      6. distribute-lft-neg-out 97.3%

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

      7. remove-double-neg 97.3%

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

      8. *-commutative 97.3%

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

      9. hypot-undefine 88.0%

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

      10. +-commutative 88.0%

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

      11. hypot-define 97.3%

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

      12. hypot-undefine 88.0%

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

      13. +-commutative 88.0%

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

      14. hypot-define 97.3%

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

   6. Applied egg-rr 97.3%

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

   if 2.00000000000000013e265 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 12.0%

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

      1. div-sub 7.5%

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

      2. *-commutative 7.5%

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

      3. add-sqr-sqrt 7.5%

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

      4. times-frac 10.3%

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

      5. fma-neg 10.3%

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

      6. hypot-define 10.3%

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

      7. hypot-define 46.7%

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

      8. associate-/l* 56.8%

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

      9. add-sqr-sqrt 56.8%

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

      10. pow2 56.8%

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

      11. hypot-define 56.8%

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

   4. Applied egg-rr 56.8%

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

      1. *-un-lft-identity 56.8%

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

      2. unpow2 56.8%

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

      3. times-frac 98.6%

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

      4. hypot-undefine 56.8%

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

      5. +-commutative 56.8%

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

      6. hypot-define 98.6%

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

      7. hypot-undefine 56.8%

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

      8. +-commutative 56.8%

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

      9. hypot-define 98.6%

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

   6. Applied egg-rr 98.6%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+265}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-\frac{a \cdot d}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \left(\frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{-1}{\mathsf{hypot}\left(d, c\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{+78} \lor \neg \left(d \leq 5 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b \cdot \frac{c}{d} - a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{-d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1e+78) (not (<= d 5e+137)))
   (* (/ d (hypot d c)) (/ (- (* b (/ c d)) a) (hypot d c)))
   (fma
    (/ c (hypot c d))
    (/ b (hypot c d))
    (* a (/ (- d) (pow (hypot c d) 2.0))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1e+78) || !(d <= 5e+137)) {
		tmp = (d / hypot(d, c)) * (((b * (c / d)) - a) / hypot(d, c));
	} else {
		tmp = fma((c / hypot(c, d)), (b / hypot(c, d)), (a * (-d / pow(hypot(c, d), 2.0))));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1e+78) || !(d <= 5e+137))
		tmp = Float64(Float64(d / hypot(d, c)) * Float64(Float64(Float64(b * Float64(c / d)) - a) / hypot(d, c)));
	else
		tmp = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(a * Float64(Float64(-d) / (hypot(c, d) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1e+78], N[Not[LessEqual[d, 5e+137]], $MachinePrecision]], N[(N[(d / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[(a * N[((-d) / N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.00000000000000001e78 or 5.0000000000000002e137 < d

   1. Initial program 35.1%

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

   3. Taylor expanded in d around inf 35.1%

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

      1. associate-/l* 33.0%

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

   5. Simplified 33.0%

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

      1. *-commutative 33.0%

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

      2. +-commutative 33.0%

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

      3. add-sqr-sqrt 33.0%

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

      4. hypot-undefine 33.0%

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

      5. hypot-undefine 33.0%

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

      6. times-frac 97.7%

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

   7. Applied egg-rr 97.7%

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

   if -1.00000000000000001e78 < d < 5.0000000000000002e137

   1. Initial program 75.0%

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

      1. div-sub 73.1%

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

      2. *-commutative 73.1%

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

      3. add-sqr-sqrt 73.1%

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

      4. times-frac 74.8%

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

      5. fma-neg 74.8%

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

      6. hypot-define 74.8%

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

      7. hypot-define 93.8%

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

      8. associate-/l* 92.1%

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

      9. add-sqr-sqrt 92.1%

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

      10. pow2 92.1%

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

      11. hypot-define 92.1%

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

   4. Applied egg-rr 92.1%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{+78} \lor \neg \left(d \leq 5 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b \cdot \frac{c}{d} - a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{-d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{c}{\mathsf{hypot}\left(d, c\right)}}{\frac{\mathsf{hypot}\left(d, c\right)}{b}}\\ \mathbf{elif}\;c \leq -3.6 \cdot 10^{-65}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.16 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 7.8 \cdot 10^{+119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, c, a \cdot \left(-d\right)\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.1e+136)
   (/ (/ c (hypot d c)) (/ (hypot d c) b))
   (if (<= c -3.6e-65)
     (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))
     (if (<= c 1.16e-80)
       (/ (- (/ (* b c) d) a) d)
       (if (<= c 7.8e+119)
         (/ (fma b c (* a (- d))) (fma d d (* c c)))
         (/ (- b (/ d (/ c a))) c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.1e+136) {
		tmp = (c / hypot(d, c)) / (hypot(d, c) / b);
	} else if (c <= -3.6e-65) {
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
	} else if (c <= 1.16e-80) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 7.8e+119) {
		tmp = fma(b, c, (a * -d)) / fma(d, d, (c * c));
	} else {
		tmp = (b - (d / (c / a))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.1e+136)
		tmp = Float64(Float64(c / hypot(d, c)) / Float64(hypot(d, c) / b));
	elseif (c <= -3.6e-65)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 1.16e-80)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 7.8e+119)
		tmp = Float64(fma(b, c, Float64(a * Float64(-d))) / fma(d, d, Float64(c * c)));
	else
		tmp = Float64(Float64(b - Float64(d / Float64(c / a))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.1e+136], N[(N[(c / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -3.6e-65], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.16e-80], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 7.8e+119], N[(N[(b * c + N[(a * (-d)), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - N[(d / N[(c / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -1.1e136

   1. Initial program 46.0%

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

   3. Taylor expanded in b around inf 46.2%

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

      1. rem-square-sqrt 46.2%

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

      2. unpow2 46.2%

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

      3. unpow2 46.2%

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

      4. hypot-undefine 46.2%

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

      5. unpow2 46.2%

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

      6. unpow2 46.2%

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

      7. hypot-undefine 46.2%

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

      8. unpow2 46.2%

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

      9. *-commutative 46.2%

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

      10. associate-*r/ 47.4%

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

      11. hypot-undefine 47.4%

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

      12. unpow2 47.4%

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

      13. unpow2 47.4%

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

      14. +-commutative 47.4%

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

      15. unpow2 47.4%

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

      16. unpow2 47.4%

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

      17. hypot-define 47.4%

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

   5. Simplified 47.4%

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

      1. associate-*r/ 46.2%

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

      2. unpow2 46.2%

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

      3. frac-times 90.1%

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

      4. hypot-undefine 47.4%

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

      5. +-commutative 47.4%

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

      6. hypot-undefine 90.1%

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

      7. hypot-undefine 47.4%

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

      8. +-commutative 47.4%

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

      9. hypot-undefine 90.1%

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

      10. clear-num 89.8%

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

      11. un-div-inv 89.8%

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

      12. hypot-undefine 47.4%

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

      13. +-commutative 47.4%

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

      14. hypot-undefine 89.8%

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

      15. hypot-undefine 47.4%

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

      16. +-commutative 47.4%

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

      17. hypot-undefine 89.8%

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

   7. Applied egg-rr 89.8%

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

   if -1.1e136 < c < -3.5999999999999998e-65

   1. Initial program 72.7%

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

   if -3.5999999999999998e-65 < c < 1.15999999999999996e-80

   1. Initial program 68.5%

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

      1. div-sub 64.9%

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

      2. *-commutative 64.9%

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

      3. add-sqr-sqrt 64.9%

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

      4. times-frac 60.8%

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

      5. fma-neg 60.8%

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

      6. hypot-define 60.8%

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

      7. hypot-define 64.2%

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

      8. associate-/l* 67.9%

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

      9. add-sqr-sqrt 67.9%

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

      10. pow2 67.9%

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

      11. hypot-define 67.9%

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

   4. Applied egg-rr 67.9%

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

   5. Taylor expanded in d around inf 93.8%

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

   if 1.15999999999999996e-80 < c < 7.7999999999999997e119

   1. Initial program 72.6%

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

      1. fma-neg 72.6%

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

      2. distribute-rgt-neg-out 72.6%

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

      3. +-commutative 72.6%

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

      4. fma-define 72.6%

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

   3. Simplified 72.6%

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

   if 7.7999999999999997e119 < c

   1. Initial program 41.2%

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

      1. div-sub 41.2%

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

      2. *-commutative 41.2%

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

      3. add-sqr-sqrt 41.2%

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

      4. times-frac 42.3%

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

      5. fma-neg 42.3%

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

      6. hypot-define 42.3%

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

      7. hypot-define 82.6%

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

      8. associate-/l* 83.1%

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

      9. add-sqr-sqrt 83.1%

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

      10. pow2 83.1%

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

      11. hypot-define 83.1%

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

   4. Applied egg-rr 83.1%

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

   5. Taylor expanded in c around inf 87.9%

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

      1. mul-1-neg 87.9%

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

      2. *-commutative 87.9%

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

      3. associate-*r/ 94.5%

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

      4. unsub-neg 94.5%

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

   7. Simplified 94.5%

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

      1. clear-num 94.5%

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

      2. un-div-inv 94.5%

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

   9. Applied egg-rr 94.5%

      \[\leadsto \frac{b - \color{blue}{\frac{d}{\frac{c}{a}}}}{c} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 5: 90.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{-82} \lor \neg \left(d \leq 2.8 \cdot 10^{-108}\right):\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b \cdot \frac{c}{d} - a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.1e-82) (not (<= d 2.8e-108)))
   (* (/ d (hypot d c)) (/ (- (* b (/ c d)) a) (hypot d c)))
   (/ (- b (/ (* a d) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.1e-82) || !(d <= 2.8e-108)) {
		tmp = (d / hypot(d, c)) * (((b * (c / d)) - a) / hypot(d, c));
	} else {
		tmp = (b - ((a * d) / c)) / c;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.1e-82) || !(d <= 2.8e-108)) {
		tmp = (d / Math.hypot(d, c)) * (((b * (c / d)) - a) / Math.hypot(d, c));
	} else {
		tmp = (b - ((a * d) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2.1e-82) or not (d <= 2.8e-108):
		tmp = (d / math.hypot(d, c)) * (((b * (c / d)) - a) / math.hypot(d, c))
	else:
		tmp = (b - ((a * d) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.1e-82) || !(d <= 2.8e-108))
		tmp = Float64(Float64(d / hypot(d, c)) * Float64(Float64(Float64(b * Float64(c / d)) - a) / hypot(d, c)));
	else
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -2.1e-82) || ~((d <= 2.8e-108)))
		tmp = (d / hypot(d, c)) * (((b * (c / d)) - a) / hypot(d, c));
	else
		tmp = (b - ((a * d) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.1e-82], N[Not[LessEqual[d, 2.8e-108]], $MachinePrecision]], N[(N[(d / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.1e-82 or 2.8e-108 < d

   1. Initial program 55.8%

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

   3. Taylor expanded in d around inf 55.3%

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

      1. associate-/l* 53.6%

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

   5. Simplified 53.6%

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

      1. *-commutative 53.6%

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

      2. +-commutative 53.6%

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

      3. add-sqr-sqrt 53.6%

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

      4. hypot-undefine 53.6%

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

      5. hypot-undefine 53.6%

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

      6. times-frac 93.2%

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

   7. Applied egg-rr 93.2%

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

   if -2.1e-82 < d < 2.8e-108

   1. Initial program 73.3%

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

   3. Taylor expanded in c around inf 95.2%

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

      1. remove-double-neg 95.2%

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

      2. mul-1-neg 95.2%

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

      3. neg-mul-1 95.2%

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

      4. distribute-lft-in 95.2%

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

      5. distribute-lft-in 95.2%

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

      6. mul-1-neg 95.2%

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

      7. unsub-neg 95.2%

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

      8. neg-mul-1 95.2%

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

      9. mul-1-neg 95.2%

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

      10. remove-double-neg 95.2%

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

      11. associate-/l* 93.3%

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

   5. Simplified 93.3%

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

      1. *-commutative 93.3%

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

      2. associate-*l/ 95.2%

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

   7. Applied egg-rr 95.2%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{-82} \lor \neg \left(d \leq 2.8 \cdot 10^{-108}\right):\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b \cdot \frac{c}{d} - a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.3 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{c}{\mathsf{hypot}\left(d, c\right)}}{\frac{\mathsf{hypot}\left(d, c\right)}{b}}\\ \mathbf{elif}\;c \leq -6.4 \cdot 10^{-69}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{+120}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))))
   (if (<= c -1.3e+136)
     (/ (/ c (hypot d c)) (/ (hypot d c) b))
     (if (<= c -6.4e-69)
       t_0
       (if (<= c 6.2e-81)
         (/ (- (/ (* b c) d) a) d)
         (if (<= c 1.6e+120) t_0 (/ (- b (/ d (/ c a))) c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1.3e+136) {
		tmp = (c / hypot(d, c)) / (hypot(d, c) / b);
	} else if (c <= -6.4e-69) {
		tmp = t_0;
	} else if (c <= 6.2e-81) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 1.6e+120) {
		tmp = t_0;
	} else {
		tmp = (b - (d / (c / a))) / c;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1.3e+136) {
		tmp = (c / Math.hypot(d, c)) / (Math.hypot(d, c) / b);
	} else if (c <= -6.4e-69) {
		tmp = t_0;
	} else if (c <= 6.2e-81) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 1.6e+120) {
		tmp = t_0;
	} else {
		tmp = (b - (d / (c / a))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -1.3e+136:
		tmp = (c / math.hypot(d, c)) / (math.hypot(d, c) / b)
	elif c <= -6.4e-69:
		tmp = t_0
	elif c <= 6.2e-81:
		tmp = (((b * c) / d) - a) / d
	elif c <= 1.6e+120:
		tmp = t_0
	else:
		tmp = (b - (d / (c / a))) / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -1.3e+136)
		tmp = Float64(Float64(c / hypot(d, c)) / Float64(hypot(d, c) / b));
	elseif (c <= -6.4e-69)
		tmp = t_0;
	elseif (c <= 6.2e-81)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 1.6e+120)
		tmp = t_0;
	else
		tmp = Float64(Float64(b - Float64(d / Float64(c / a))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -1.3e+136)
		tmp = (c / hypot(d, c)) / (hypot(d, c) / b);
	elseif (c <= -6.4e-69)
		tmp = t_0;
	elseif (c <= 6.2e-81)
		tmp = (((b * c) / d) - a) / d;
	elseif (c <= 1.6e+120)
		tmp = t_0;
	else
		tmp = (b - (d / (c / a))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.3e+136], N[(N[(c / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -6.4e-69], t$95$0, If[LessEqual[c, 6.2e-81], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.6e+120], t$95$0, N[(N[(b - N[(d / N[(c / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.3000000000000001e136

   1. Initial program 46.0%

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

   3. Taylor expanded in b around inf 46.2%

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

      1. rem-square-sqrt 46.2%

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

      2. unpow2 46.2%

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

      3. unpow2 46.2%

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

      4. hypot-undefine 46.2%

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

      5. unpow2 46.2%

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

      6. unpow2 46.2%

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

      7. hypot-undefine 46.2%

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

      8. unpow2 46.2%

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

      9. *-commutative 46.2%

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

      10. associate-*r/ 47.4%

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

      11. hypot-undefine 47.4%

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

      12. unpow2 47.4%

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

      13. unpow2 47.4%

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

      14. +-commutative 47.4%

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

      15. unpow2 47.4%

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

      16. unpow2 47.4%

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

      17. hypot-define 47.4%

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

   5. Simplified 47.4%

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

      1. associate-*r/ 46.2%

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

      2. unpow2 46.2%

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

      3. frac-times 90.1%

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

      4. hypot-undefine 47.4%

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

      5. +-commutative 47.4%

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

      6. hypot-undefine 90.1%

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

      7. hypot-undefine 47.4%

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

      8. +-commutative 47.4%

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

      9. hypot-undefine 90.1%

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

      10. clear-num 89.8%

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

      11. un-div-inv 89.8%

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

      12. hypot-undefine 47.4%

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

      13. +-commutative 47.4%

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

      14. hypot-undefine 89.8%

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

      15. hypot-undefine 47.4%

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

      16. +-commutative 47.4%

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

      17. hypot-undefine 89.8%

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

   7. Applied egg-rr 89.8%

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

   if -1.3000000000000001e136 < c < -6.39999999999999997e-69 or 6.19999999999999976e-81 < c < 1.59999999999999991e120

   1. Initial program 72.6%

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

   if -6.39999999999999997e-69 < c < 6.19999999999999976e-81

   1. Initial program 68.5%

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

      1. div-sub 64.9%

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

      2. *-commutative 64.9%

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

      3. add-sqr-sqrt 64.9%

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

      4. times-frac 60.8%

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

      5. fma-neg 60.8%

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

      6. hypot-define 60.8%

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

      7. hypot-define 64.2%

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

      8. associate-/l* 67.9%

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

      9. add-sqr-sqrt 67.9%

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

      10. pow2 67.9%

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

      11. hypot-define 67.9%

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

   4. Applied egg-rr 67.9%

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

   5. Taylor expanded in d around inf 93.8%

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

   if 1.59999999999999991e120 < c

   1. Initial program 41.2%

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

      1. div-sub 41.2%

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

      2. *-commutative 41.2%

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

      3. add-sqr-sqrt 41.2%

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

      4. times-frac 42.3%

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

      5. fma-neg 42.3%

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

      6. hypot-define 42.3%

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

      7. hypot-define 82.6%

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

      8. associate-/l* 83.1%

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

      9. add-sqr-sqrt 83.1%

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

      10. pow2 83.1%

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

      11. hypot-define 83.1%

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

   4. Applied egg-rr 83.1%

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

   5. Taylor expanded in c around inf 87.9%

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

      1. mul-1-neg 87.9%

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

      2. *-commutative 87.9%

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

      3. associate-*r/ 94.5%

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

      4. unsub-neg 94.5%

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

   7. Simplified 94.5%

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

      1. clear-num 94.5%

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

      2. un-div-inv 94.5%

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

   9. Applied egg-rr 94.5%

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

Alternative 7: 81.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -9 \cdot 10^{+83}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-84}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-168}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{+130}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))))
   (if (<= d -9e+83)
     (/ (- (* c (/ b d)) a) d)
     (if (<= d -7.2e-84)
       t_0
       (if (<= d 8e-168)
         (/ (- b (/ (* a d) c)) c)
         (if (<= d 6.2e+130) t_0 (/ (- (* b (/ c d)) a) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -9e+83) {
		tmp = ((c * (b / d)) - a) / d;
	} else if (d <= -7.2e-84) {
		tmp = t_0;
	} else if (d <= 8e-168) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 6.2e+130) {
		tmp = t_0;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
    if (d <= (-9d+83)) then
        tmp = ((c * (b / d)) - a) / d
    else if (d <= (-7.2d-84)) then
        tmp = t_0
    else if (d <= 8d-168) then
        tmp = (b - ((a * d) / c)) / c
    else if (d <= 6.2d+130) then
        tmp = t_0
    else
        tmp = ((b * (c / d)) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -9e+83) {
		tmp = ((c * (b / d)) - a) / d;
	} else if (d <= -7.2e-84) {
		tmp = t_0;
	} else if (d <= 8e-168) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 6.2e+130) {
		tmp = t_0;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -9e+83:
		tmp = ((c * (b / d)) - a) / d
	elif d <= -7.2e-84:
		tmp = t_0
	elif d <= 8e-168:
		tmp = (b - ((a * d) / c)) / c
	elif d <= 6.2e+130:
		tmp = t_0
	else:
		tmp = ((b * (c / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -9e+83)
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	elseif (d <= -7.2e-84)
		tmp = t_0;
	elseif (d <= 8e-168)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 6.2e+130)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -9e+83)
		tmp = ((c * (b / d)) - a) / d;
	elseif (d <= -7.2e-84)
		tmp = t_0;
	elseif (d <= 8e-168)
		tmp = (b - ((a * d) / c)) / c;
	elseif (d <= 6.2e+130)
		tmp = t_0;
	else
		tmp = ((b * (c / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -9e+83], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -7.2e-84], t$95$0, If[LessEqual[d, 8e-168], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 6.2e+130], t$95$0, N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -8.9999999999999999e83

   1. Initial program 40.9%

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

   3. Taylor expanded in c around 0 79.9%

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

      1. +-commutative 79.9%

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

      2. mul-1-neg 79.9%

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

      3. unsub-neg 79.9%

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

      4. unpow2 79.9%

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

      5. associate-/r* 80.1%

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

      6. div-sub 80.1%

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

      7. associate-/l* 82.4%

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

   5. Simplified 82.4%

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

      1. clear-num 82.4%

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

      2. un-div-inv 82.4%

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

   7. Applied egg-rr 82.4%

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

      1. associate-/r/ 86.3%

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

   9. Simplified 86.3%

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

   if -8.9999999999999999e83 < d < -7.20000000000000007e-84 or 8.0000000000000004e-168 < d < 6.1999999999999999e130

   1. Initial program 79.8%

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

   if -7.20000000000000007e-84 < d < 8.0000000000000004e-168

   1. Initial program 70.8%

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

   3. Taylor expanded in c around inf 95.9%

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

      1. remove-double-neg 95.9%

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

      2. mul-1-neg 95.9%

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

      3. neg-mul-1 95.9%

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

      4. distribute-lft-in 95.9%

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

      5. distribute-lft-in 95.9%

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

      6. mul-1-neg 95.9%

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

      7. unsub-neg 95.9%

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

      8. neg-mul-1 95.9%

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

      9. mul-1-neg 95.9%

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

      10. remove-double-neg 95.9%

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

      11. associate-/l* 93.8%

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

   5. Simplified 93.8%

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

      1. *-commutative 93.8%

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

      2. associate-*l/ 95.9%

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

   7. Applied egg-rr 95.9%

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

   if 6.1999999999999999e130 < d

   1. Initial program 25.5%

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

   3. Taylor expanded in c around 0 70.9%

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

      1. +-commutative 70.9%

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

      2. mul-1-neg 70.9%

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

      3. unsub-neg 70.9%

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

      4. unpow2 70.9%

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

      5. associate-/r* 78.6%

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

      6. div-sub 78.6%

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

      7. associate-/l* 83.4%

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

   5. Simplified 83.4%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{+83}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-168}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{+130}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \mathbf{if}\;c \leq -1.2 \cdot 10^{+60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -6.5 \cdot 10^{-49}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{elif}\;c \leq -2.4 \cdot 10^{-64}:\\ \;\;\;\;\frac{1}{\frac{c}{b - a \cdot \frac{d}{c}}}\\ \mathbf{elif}\;c \leq 50000:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (/ d (/ c a))) c)))
   (if (<= c -1.2e+60)
     t_0
     (if (<= c -6.5e-49)
       (/ (- (/ b (/ d c)) a) d)
       (if (<= c -2.4e-64)
         (/ 1.0 (/ c (- b (* a (/ d c)))))
         (if (<= c 50000.0) (/ (- (/ (* b c) d) a) d) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - (d / (c / a))) / c;
	double tmp;
	if (c <= -1.2e+60) {
		tmp = t_0;
	} else if (c <= -6.5e-49) {
		tmp = ((b / (d / c)) - a) / d;
	} else if (c <= -2.4e-64) {
		tmp = 1.0 / (c / (b - (a * (d / c))));
	} else if (c <= 50000.0) {
		tmp = (((b * c) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b - (d / (c / a))) / c
    if (c <= (-1.2d+60)) then
        tmp = t_0
    else if (c <= (-6.5d-49)) then
        tmp = ((b / (d / c)) - a) / d
    else if (c <= (-2.4d-64)) then
        tmp = 1.0d0 / (c / (b - (a * (d / c))))
    else if (c <= 50000.0d0) then
        tmp = (((b * c) / d) - a) / d
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b - (d / (c / a))) / c;
	double tmp;
	if (c <= -1.2e+60) {
		tmp = t_0;
	} else if (c <= -6.5e-49) {
		tmp = ((b / (d / c)) - a) / d;
	} else if (c <= -2.4e-64) {
		tmp = 1.0 / (c / (b - (a * (d / c))));
	} else if (c <= 50000.0) {
		tmp = (((b * c) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b - (d / (c / a))) / c
	tmp = 0
	if c <= -1.2e+60:
		tmp = t_0
	elif c <= -6.5e-49:
		tmp = ((b / (d / c)) - a) / d
	elif c <= -2.4e-64:
		tmp = 1.0 / (c / (b - (a * (d / c))))
	elif c <= 50000.0:
		tmp = (((b * c) / d) - a) / d
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(d / Float64(c / a))) / c)
	tmp = 0.0
	if (c <= -1.2e+60)
		tmp = t_0;
	elseif (c <= -6.5e-49)
		tmp = Float64(Float64(Float64(b / Float64(d / c)) - a) / d);
	elseif (c <= -2.4e-64)
		tmp = Float64(1.0 / Float64(c / Float64(b - Float64(a * Float64(d / c)))));
	elseif (c <= 50000.0)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b - (d / (c / a))) / c;
	tmp = 0.0;
	if (c <= -1.2e+60)
		tmp = t_0;
	elseif (c <= -6.5e-49)
		tmp = ((b / (d / c)) - a) / d;
	elseif (c <= -2.4e-64)
		tmp = 1.0 / (c / (b - (a * (d / c))));
	elseif (c <= 50000.0)
		tmp = (((b * c) / d) - a) / d;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(d / N[(c / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -1.2e+60], t$95$0, If[LessEqual[c, -6.5e-49], N[(N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, -2.4e-64], N[(1.0 / N[(c / N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 50000.0], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.2e60 or 5e4 < c

   1. Initial program 53.6%

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

      1. div-sub 53.6%

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

      2. *-commutative 53.6%

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

      3. add-sqr-sqrt 53.6%

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

      4. times-frac 57.4%

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

      5. fma-neg 57.4%

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

      6. hypot-define 57.4%

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

      7. hypot-define 87.2%

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

      8. associate-/l* 86.8%

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

      9. add-sqr-sqrt 86.8%

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

      10. pow2 86.8%

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

      11. hypot-define 86.8%

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

   4. Applied egg-rr 86.8%

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

   5. Taylor expanded in c around inf 78.7%

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

      1. mul-1-neg 78.7%

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

      2. *-commutative 78.7%

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

      3. associate-*r/ 83.1%

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

      4. unsub-neg 83.1%

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

   7. Simplified 83.1%

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

      1. clear-num 83.1%

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

      2. un-div-inv 83.1%

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

   9. Applied egg-rr 83.1%

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

   if -1.2e60 < c < -6.49999999999999968e-49

   1. Initial program 56.1%

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

   3. Taylor expanded in c around 0 57.0%

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

      1. +-commutative 57.0%

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

      2. mul-1-neg 57.0%

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

      3. unsub-neg 57.0%

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

      4. unpow2 57.0%

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

      5. associate-/r* 62.5%

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

      6. div-sub 62.5%

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

      7. associate-/l* 66.7%

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

   5. Simplified 66.7%

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

      1. clear-num 66.7%

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

      2. un-div-inv 66.7%

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

   7. Applied egg-rr 66.7%

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

   if -6.49999999999999968e-49 < c < -2.39999999999999998e-64

   1. Initial program 99.4%

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

      1. div-sub 99.4%

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

      2. *-commutative 99.4%

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

      3. add-sqr-sqrt 99.4%

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

      4. times-frac 99.4%

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

      5. fma-neg 99.4%

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

      6. hypot-define 99.4%

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

      7. hypot-define 99.4%

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

      8. associate-/l* 99.4%

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

      9. add-sqr-sqrt 99.4%

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

      10. pow2 99.4%

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

      11. hypot-define 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in c around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*r/ 42.8%

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

      4. unsub-neg 42.8%

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

   7. Simplified 42.8%

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

      1. clear-num 42.5%

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

      2. inv-pow 42.5%

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

      3. associate-*r/ 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. unpow-1 100.0%

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

       2. *-commutative 100.0%

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

       3. associate-/l* 100.0%

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

   11. Simplified 100.0%

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

   if -2.39999999999999998e-64 < c < 5e4

   1. Initial program 70.2%

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

      1. div-sub 67.1%

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

      2. *-commutative 67.1%

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

      3. add-sqr-sqrt 67.1%

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

      4. times-frac 63.4%

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

      5. fma-neg 63.4%

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

      6. hypot-define 63.4%

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

      7. hypot-define 66.4%

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

      8. associate-/l* 69.8%

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

      9. add-sqr-sqrt 69.8%

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

      10. pow2 69.8%

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

      11. hypot-define 69.8%

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

   4. Applied egg-rr 69.8%

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

   5. Taylor expanded in d around inf 90.1%

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

Alternative 9: 77.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \mathbf{if}\;c \leq -7.8 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -2.3 \cdot 10^{-47}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{elif}\;c \leq -4 \cdot 10^{-65}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;c \leq 2900000:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (/ d (/ c a))) c)))
   (if (<= c -7.8e+59)
     t_0
     (if (<= c -2.3e-47)
       (/ (- (/ b (/ d c)) a) d)
       (if (<= c -4e-65)
         (/ (- b (/ (* a d) c)) c)
         (if (<= c 2900000.0) (/ (- (/ (* b c) d) a) d) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - (d / (c / a))) / c;
	double tmp;
	if (c <= -7.8e+59) {
		tmp = t_0;
	} else if (c <= -2.3e-47) {
		tmp = ((b / (d / c)) - a) / d;
	} else if (c <= -4e-65) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (c <= 2900000.0) {
		tmp = (((b * c) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b - (d / (c / a))) / c
    if (c <= (-7.8d+59)) then
        tmp = t_0
    else if (c <= (-2.3d-47)) then
        tmp = ((b / (d / c)) - a) / d
    else if (c <= (-4d-65)) then
        tmp = (b - ((a * d) / c)) / c
    else if (c <= 2900000.0d0) then
        tmp = (((b * c) / d) - a) / d
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b - (d / (c / a))) / c;
	double tmp;
	if (c <= -7.8e+59) {
		tmp = t_0;
	} else if (c <= -2.3e-47) {
		tmp = ((b / (d / c)) - a) / d;
	} else if (c <= -4e-65) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (c <= 2900000.0) {
		tmp = (((b * c) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b - (d / (c / a))) / c
	tmp = 0
	if c <= -7.8e+59:
		tmp = t_0
	elif c <= -2.3e-47:
		tmp = ((b / (d / c)) - a) / d
	elif c <= -4e-65:
		tmp = (b - ((a * d) / c)) / c
	elif c <= 2900000.0:
		tmp = (((b * c) / d) - a) / d
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(d / Float64(c / a))) / c)
	tmp = 0.0
	if (c <= -7.8e+59)
		tmp = t_0;
	elseif (c <= -2.3e-47)
		tmp = Float64(Float64(Float64(b / Float64(d / c)) - a) / d);
	elseif (c <= -4e-65)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (c <= 2900000.0)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b - (d / (c / a))) / c;
	tmp = 0.0;
	if (c <= -7.8e+59)
		tmp = t_0;
	elseif (c <= -2.3e-47)
		tmp = ((b / (d / c)) - a) / d;
	elseif (c <= -4e-65)
		tmp = (b - ((a * d) / c)) / c;
	elseif (c <= 2900000.0)
		tmp = (((b * c) / d) - a) / d;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(d / N[(c / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -7.8e+59], t$95$0, If[LessEqual[c, -2.3e-47], N[(N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, -4e-65], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 2900000.0], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -7.80000000000000043e59 or 2.9e6 < c

   1. Initial program 53.6%

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

      1. div-sub 53.6%

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

      2. *-commutative 53.6%

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

      3. add-sqr-sqrt 53.6%

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

      4. times-frac 57.4%

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

      5. fma-neg 57.4%

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

      6. hypot-define 57.4%

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

      7. hypot-define 87.2%

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

      8. associate-/l* 86.8%

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

      9. add-sqr-sqrt 86.8%

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

      10. pow2 86.8%

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

      11. hypot-define 86.8%

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

   4. Applied egg-rr 86.8%

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

   5. Taylor expanded in c around inf 78.7%

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

      1. mul-1-neg 78.7%

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

      2. *-commutative 78.7%

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

      3. associate-*r/ 83.1%

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

      4. unsub-neg 83.1%

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

   7. Simplified 83.1%

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

      1. clear-num 83.1%

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

      2. un-div-inv 83.1%

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

   9. Applied egg-rr 83.1%

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

   if -7.80000000000000043e59 < c < -2.29999999999999982e-47

   1. Initial program 56.1%

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

   3. Taylor expanded in c around 0 57.0%

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

      1. +-commutative 57.0%

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

      2. mul-1-neg 57.0%

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

      3. unsub-neg 57.0%

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

      4. unpow2 57.0%

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

      5. associate-/r* 62.5%

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

      6. div-sub 62.5%

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

      7. associate-/l* 66.7%

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

   5. Simplified 66.7%

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

      1. clear-num 66.7%

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

      2. un-div-inv 66.7%

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

   7. Applied egg-rr 66.7%

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

   if -2.29999999999999982e-47 < c < -3.99999999999999969e-65

   1. Initial program 99.4%

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

   3. Taylor expanded in c around inf 99.7%

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

      1. remove-double-neg 99.7%

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

      2. mul-1-neg 99.7%

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

      3. neg-mul-1 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. distribute-lft-in 99.7%

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

      6. mul-1-neg 99.7%

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

      7. unsub-neg 99.7%

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

      8. neg-mul-1 99.7%

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

      9. mul-1-neg 99.7%

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

      10. remove-double-neg 99.7%

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

      11. associate-/l* 99.4%

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

   5. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l/ 99.7%

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

   7. Applied egg-rr 99.7%

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

   if -3.99999999999999969e-65 < c < 2.9e6

   1. Initial program 70.2%

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

      1. div-sub 67.1%

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

      2. *-commutative 67.1%

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

      3. add-sqr-sqrt 67.1%

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

      4. times-frac 63.4%

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

      5. fma-neg 63.4%

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

      6. hypot-define 63.4%

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

      7. hypot-define 66.4%

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

      8. associate-/l* 69.8%

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

      9. add-sqr-sqrt 69.8%

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

      10. pow2 69.8%

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

      11. hypot-define 69.8%

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

   4. Applied egg-rr 69.8%

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

   5. Taylor expanded in d around inf 90.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.8 \cdot 10^{+59}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \mathbf{elif}\;c \leq -2.3 \cdot 10^{-47}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{elif}\;c \leq -4 \cdot 10^{-65}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;c \leq 2900000:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{b}{\frac{d}{c}} - a}{d}\\ t_1 := \frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \mathbf{if}\;c \leq -1.1 \cdot 10^{+66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -1.65 \cdot 10^{-45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -2.4 \cdot 10^{-64}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;c \leq 16000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (/ b (/ d c)) a) d)) (t_1 (/ (- b (/ d (/ c a))) c)))
   (if (<= c -1.1e+66)
     t_1
     (if (<= c -1.65e-45)
       t_0
       (if (<= c -2.4e-64)
         (/ (- b (/ (* a d) c)) c)
         (if (<= c 16000.0) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b / (d / c)) - a) / d;
	double t_1 = (b - (d / (c / a))) / c;
	double tmp;
	if (c <= -1.1e+66) {
		tmp = t_1;
	} else if (c <= -1.65e-45) {
		tmp = t_0;
	} else if (c <= -2.4e-64) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (c <= 16000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((b / (d / c)) - a) / d
    t_1 = (b - (d / (c / a))) / c
    if (c <= (-1.1d+66)) then
        tmp = t_1
    else if (c <= (-1.65d-45)) then
        tmp = t_0
    else if (c <= (-2.4d-64)) then
        tmp = (b - ((a * d) / c)) / c
    else if (c <= 16000.0d0) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b / (d / c)) - a) / d;
	double t_1 = (b - (d / (c / a))) / c;
	double tmp;
	if (c <= -1.1e+66) {
		tmp = t_1;
	} else if (c <= -1.65e-45) {
		tmp = t_0;
	} else if (c <= -2.4e-64) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (c <= 16000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b / (d / c)) - a) / d
	t_1 = (b - (d / (c / a))) / c
	tmp = 0
	if c <= -1.1e+66:
		tmp = t_1
	elif c <= -1.65e-45:
		tmp = t_0
	elif c <= -2.4e-64:
		tmp = (b - ((a * d) / c)) / c
	elif c <= 16000.0:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b / Float64(d / c)) - a) / d)
	t_1 = Float64(Float64(b - Float64(d / Float64(c / a))) / c)
	tmp = 0.0
	if (c <= -1.1e+66)
		tmp = t_1;
	elseif (c <= -1.65e-45)
		tmp = t_0;
	elseif (c <= -2.4e-64)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (c <= 16000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b / (d / c)) - a) / d;
	t_1 = (b - (d / (c / a))) / c;
	tmp = 0.0;
	if (c <= -1.1e+66)
		tmp = t_1;
	elseif (c <= -1.65e-45)
		tmp = t_0;
	elseif (c <= -2.4e-64)
		tmp = (b - ((a * d) / c)) / c;
	elseif (c <= 16000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b - N[(d / N[(c / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -1.1e+66], t$95$1, If[LessEqual[c, -1.65e-45], t$95$0, If[LessEqual[c, -2.4e-64], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 16000.0], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.0999999999999999e66 or 16000 < c

   1. Initial program 53.6%

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

      1. div-sub 53.6%

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

      2. *-commutative 53.6%

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

      3. add-sqr-sqrt 53.6%

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

      4. times-frac 57.4%

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

      5. fma-neg 57.4%

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

      6. hypot-define 57.4%

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

      7. hypot-define 87.2%

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

      8. associate-/l* 86.8%

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

      9. add-sqr-sqrt 86.8%

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

      10. pow2 86.8%

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

      11. hypot-define 86.8%

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

   4. Applied egg-rr 86.8%

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

   5. Taylor expanded in c around inf 78.7%

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

      1. mul-1-neg 78.7%

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

      2. *-commutative 78.7%

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

      3. associate-*r/ 83.1%

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

      4. unsub-neg 83.1%

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

   7. Simplified 83.1%

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

      1. clear-num 83.1%

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

      2. un-div-inv 83.1%

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

   9. Applied egg-rr 83.1%

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

   if -1.0999999999999999e66 < c < -1.65e-45 or -2.39999999999999998e-64 < c < 16000

   1. Initial program 67.6%

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

   3. Taylor expanded in c around 0 79.8%

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

      1. +-commutative 79.8%

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

      2. mul-1-neg 79.8%

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

      3. unsub-neg 79.8%

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

      4. unpow2 79.8%

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

      5. associate-/r* 85.0%

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

      6. div-sub 85.0%

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

      7. associate-/l* 85.0%

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

   5. Simplified 85.0%

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

      1. clear-num 85.0%

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

      2. un-div-inv 85.0%

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

   7. Applied egg-rr 85.0%

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

   if -1.65e-45 < c < -2.39999999999999998e-64

   1. Initial program 99.4%

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

   3. Taylor expanded in c around inf 99.7%

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

      1. remove-double-neg 99.7%

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

      2. mul-1-neg 99.7%

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

      3. neg-mul-1 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. distribute-lft-in 99.7%

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

      6. mul-1-neg 99.7%

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

      7. unsub-neg 99.7%

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

      8. neg-mul-1 99.7%

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

      9. mul-1-neg 99.7%

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

      10. remove-double-neg 99.7%

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

      11. associate-/l* 99.4%

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

   5. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l/ 99.7%

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

   7. Applied egg-rr 99.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{+66}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \mathbf{elif}\;c \leq -1.65 \cdot 10^{-45}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{elif}\;c \leq -2.4 \cdot 10^{-64}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;c \leq 16000:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot \frac{c}{d} - a}{d}\\ t_1 := \frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \mathbf{if}\;c \leq -6.8 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -1.6 \cdot 10^{-41}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -2.4 \cdot 10^{-64}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;c \leq 9000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b (/ c d)) a) d)) (t_1 (/ (- b (/ d (/ c a))) c)))
   (if (<= c -6.8e+59)
     t_1
     (if (<= c -1.6e-41)
       t_0
       (if (<= c -2.4e-64)
         (/ (- b (/ (* a d) c)) c)
         (if (<= c 9000.0) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * (c / d)) - a) / d;
	double t_1 = (b - (d / (c / a))) / c;
	double tmp;
	if (c <= -6.8e+59) {
		tmp = t_1;
	} else if (c <= -1.6e-41) {
		tmp = t_0;
	} else if (c <= -2.4e-64) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (c <= 9000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((b * (c / d)) - a) / d
    t_1 = (b - (d / (c / a))) / c
    if (c <= (-6.8d+59)) then
        tmp = t_1
    else if (c <= (-1.6d-41)) then
        tmp = t_0
    else if (c <= (-2.4d-64)) then
        tmp = (b - ((a * d) / c)) / c
    else if (c <= 9000.0d0) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * (c / d)) - a) / d;
	double t_1 = (b - (d / (c / a))) / c;
	double tmp;
	if (c <= -6.8e+59) {
		tmp = t_1;
	} else if (c <= -1.6e-41) {
		tmp = t_0;
	} else if (c <= -2.4e-64) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (c <= 9000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * (c / d)) - a) / d
	t_1 = (b - (d / (c / a))) / c
	tmp = 0
	if c <= -6.8e+59:
		tmp = t_1
	elif c <= -1.6e-41:
		tmp = t_0
	elif c <= -2.4e-64:
		tmp = (b - ((a * d) / c)) / c
	elif c <= 9000.0:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * Float64(c / d)) - a) / d)
	t_1 = Float64(Float64(b - Float64(d / Float64(c / a))) / c)
	tmp = 0.0
	if (c <= -6.8e+59)
		tmp = t_1;
	elseif (c <= -1.6e-41)
		tmp = t_0;
	elseif (c <= -2.4e-64)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (c <= 9000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * (c / d)) - a) / d;
	t_1 = (b - (d / (c / a))) / c;
	tmp = 0.0;
	if (c <= -6.8e+59)
		tmp = t_1;
	elseif (c <= -1.6e-41)
		tmp = t_0;
	elseif (c <= -2.4e-64)
		tmp = (b - ((a * d) / c)) / c;
	elseif (c <= 9000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b - N[(d / N[(c / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -6.8e+59], t$95$1, If[LessEqual[c, -1.6e-41], t$95$0, If[LessEqual[c, -2.4e-64], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 9000.0], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -6.80000000000000012e59 or 9e3 < c

   1. Initial program 53.6%

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

      1. div-sub 53.6%

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

      2. *-commutative 53.6%

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

      3. add-sqr-sqrt 53.6%

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

      4. times-frac 57.4%

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

      5. fma-neg 57.4%

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

      6. hypot-define 57.4%

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

      7. hypot-define 87.2%

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

      8. associate-/l* 86.8%

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

      9. add-sqr-sqrt 86.8%

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

      10. pow2 86.8%

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

      11. hypot-define 86.8%

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

   4. Applied egg-rr 86.8%

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

   5. Taylor expanded in c around inf 78.7%

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

      1. mul-1-neg 78.7%

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

      2. *-commutative 78.7%

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

      3. associate-*r/ 83.1%

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

      4. unsub-neg 83.1%

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

   7. Simplified 83.1%

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

      1. clear-num 83.1%

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

      2. un-div-inv 83.1%

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

   9. Applied egg-rr 83.1%

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

   if -6.80000000000000012e59 < c < -1.60000000000000006e-41 or -2.39999999999999998e-64 < c < 9e3

   1. Initial program 67.6%

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

   3. Taylor expanded in c around 0 79.8%

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

      1. +-commutative 79.8%

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

      2. mul-1-neg 79.8%

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

      3. unsub-neg 79.8%

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

      4. unpow2 79.8%

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

      5. associate-/r* 85.0%

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

      6. div-sub 85.0%

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

      7. associate-/l* 85.0%

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

   5. Simplified 85.0%

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

   if -1.60000000000000006e-41 < c < -2.39999999999999998e-64

   1. Initial program 99.4%

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

   3. Taylor expanded in c around inf 99.7%

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

      1. remove-double-neg 99.7%

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

      2. mul-1-neg 99.7%

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

      3. neg-mul-1 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. distribute-lft-in 99.7%

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

      6. mul-1-neg 99.7%

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

      7. unsub-neg 99.7%

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

      8. neg-mul-1 99.7%

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

      9. mul-1-neg 99.7%

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

      10. remove-double-neg 99.7%

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

      11. associate-/l* 99.4%

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

   5. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l/ 99.7%

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

   7. Applied egg-rr 99.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.8 \cdot 10^{+59}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \mathbf{elif}\;c \leq -1.6 \cdot 10^{-41}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq -2.4 \cdot 10^{-64}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;c \leq 9000:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{-12} \lor \neg \left(c \leq 2500\right):\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.2e-12) (not (<= c 2500.0)))
   (/ (- b (/ d (/ c a))) c)
   (/ (- a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.2e-12) || !(c <= 2500.0)) {
		tmp = (b - (d / (c / a))) / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-2.2d-12)) .or. (.not. (c <= 2500.0d0))) then
        tmp = (b - (d / (c / a))) / c
    else
        tmp = -a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.2e-12) || !(c <= 2500.0)) {
		tmp = (b - (d / (c / a))) / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.2e-12) or not (c <= 2500.0):
		tmp = (b - (d / (c / a))) / c
	else:
		tmp = -a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.2e-12) || !(c <= 2500.0))
		tmp = Float64(Float64(b - Float64(d / Float64(c / a))) / c);
	else
		tmp = Float64(Float64(-a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -2.2e-12) || ~((c <= 2500.0)))
		tmp = (b - (d / (c / a))) / c;
	else
		tmp = -a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.2e-12], N[Not[LessEqual[c, 2500.0]], $MachinePrecision]], N[(N[(b - N[(d / N[(c / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[((-a) / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.19999999999999992e-12 or 2500 < c

   1. Initial program 54.2%

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

      1. div-sub 54.2%

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

      2. *-commutative 54.2%

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

      3. add-sqr-sqrt 54.2%

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

      4. times-frac 57.6%

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

      5. fma-neg 57.6%

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

      6. hypot-define 57.6%

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

      7. hypot-define 85.1%

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

      8. associate-/l* 85.5%

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

      9. add-sqr-sqrt 85.5%

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

      10. pow2 85.5%

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

      11. hypot-define 85.5%

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

   4. Applied egg-rr 85.5%

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

   5. Taylor expanded in c around inf 75.5%

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

      1. mul-1-neg 75.5%

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

      2. *-commutative 75.5%

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

      3. associate-*r/ 79.4%

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

      4. unsub-neg 79.4%

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

   7. Simplified 79.4%

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

      1. clear-num 79.4%

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

      2. un-div-inv 79.4%

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

   9. Applied egg-rr 79.4%

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

   if -2.19999999999999992e-12 < c < 2500

   1. Initial program 70.1%

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

   3. Taylor expanded in c around 0 73.2%

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

      1. associate-*r/ 73.2%

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

      2. neg-mul-1 73.2%

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

   5. Simplified 73.2%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{-12} \lor \neg \left(c \leq 2500\right):\\ \;\;\;\;\frac{b - \frac{d}{\frac{c}{a}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.2 \cdot 10^{-13} \lor \neg \left(c \leq 6200\right):\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.2e-13) (not (<= c 6200.0)))
   (/ (- b (* d (/ a c))) c)
   (/ (- a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.2e-13) || !(c <= 6200.0)) {
		tmp = (b - (d * (a / c))) / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-1.2d-13)) .or. (.not. (c <= 6200.0d0))) then
        tmp = (b - (d * (a / c))) / c
    else
        tmp = -a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.2e-13) || !(c <= 6200.0)) {
		tmp = (b - (d * (a / c))) / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.2e-13) or not (c <= 6200.0):
		tmp = (b - (d * (a / c))) / c
	else:
		tmp = -a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.2e-13) || !(c <= 6200.0))
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	else
		tmp = Float64(Float64(-a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.2e-13) || ~((c <= 6200.0)))
		tmp = (b - (d * (a / c))) / c;
	else
		tmp = -a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.2e-13], N[Not[LessEqual[c, 6200.0]], $MachinePrecision]], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[((-a) / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.1999999999999999e-13 or 6200 < c

   1. Initial program 54.2%

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

      1. div-sub 54.2%

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

      2. *-commutative 54.2%

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

      3. add-sqr-sqrt 54.2%

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

      4. times-frac 57.6%

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

      5. fma-neg 57.6%

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

      6. hypot-define 57.6%

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

      7. hypot-define 85.1%

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

      8. associate-/l* 85.5%

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

      9. add-sqr-sqrt 85.5%

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

      10. pow2 85.5%

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

      11. hypot-define 85.5%

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

   4. Applied egg-rr 85.5%

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

   5. Taylor expanded in c around inf 75.5%

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

      1. mul-1-neg 75.5%

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

      2. *-commutative 75.5%

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

      3. associate-*r/ 79.4%

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

      4. unsub-neg 79.4%

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

   7. Simplified 79.4%

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

   if -1.1999999999999999e-13 < c < 6200

   1. Initial program 70.1%

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

   3. Taylor expanded in c around 0 73.2%

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

      1. associate-*r/ 73.2%

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

      2. neg-mul-1 73.2%

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

   5. Simplified 73.2%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.2 \cdot 10^{-13} \lor \neg \left(c \leq 6200\right):\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -53000000 \lor \neg \left(d \leq 6.5 \cdot 10^{+144}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -53000000.0) (not (<= d 6.5e+144)))
   (/ (- a) d)
   (/ (- b (* a (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -53000000.0) || !(d <= 6.5e+144)) {
		tmp = -a / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-53000000.0d0)) .or. (.not. (d <= 6.5d+144))) then
        tmp = -a / d
    else
        tmp = (b - (a * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -53000000.0) || !(d <= 6.5e+144)) {
		tmp = -a / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -53000000.0) or not (d <= 6.5e+144):
		tmp = -a / d
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -53000000.0) || !(d <= 6.5e+144))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -53000000.0) || ~((d <= 6.5e+144)))
		tmp = -a / d;
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -53000000.0], N[Not[LessEqual[d, 6.5e+144]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -5.3e7 or 6.50000000000000007e144 < d

   1. Initial program 43.8%

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

   3. Taylor expanded in c around 0 72.5%

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

      1. associate-*r/ 72.5%

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

      2. neg-mul-1 72.5%

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

   5. Simplified 72.5%

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

   if -5.3e7 < d < 6.50000000000000007e144

   1. Initial program 73.5%

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

   3. Taylor expanded in c around inf 79.2%

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

      1. remove-double-neg 79.2%

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

      2. mul-1-neg 79.2%

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

      3. neg-mul-1 79.2%

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

      4. distribute-lft-in 79.2%

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

      5. distribute-lft-in 79.2%

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

      6. mul-1-neg 79.2%

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

      7. unsub-neg 79.2%

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

      8. neg-mul-1 79.2%

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

      9. mul-1-neg 79.2%

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

      10. remove-double-neg 79.2%

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

      11. associate-/l* 78.9%

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

   5. Simplified 78.9%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -53000000 \lor \neg \left(d \leq 6.5 \cdot 10^{+144}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.2 \cdot 10^{+74} \lor \neg \left(c \leq 4600\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -5.2e+74) (not (<= c 4600.0))) (/ b c) (/ (- a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.2e+74) || !(c <= 4600.0)) {
		tmp = b / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-5.2d+74)) .or. (.not. (c <= 4600.0d0))) then
        tmp = b / c
    else
        tmp = -a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.2e+74) || !(c <= 4600.0)) {
		tmp = b / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -5.2e+74) or not (c <= 4600.0):
		tmp = b / c
	else:
		tmp = -a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -5.2e+74) || !(c <= 4600.0))
		tmp = Float64(b / c);
	else
		tmp = Float64(Float64(-a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -5.2e+74) || ~((c <= 4600.0)))
		tmp = b / c;
	else
		tmp = -a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -5.2e+74], N[Not[LessEqual[c, 4600.0]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[((-a) / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -5.2000000000000001e74 or 4600 < c

   1. Initial program 53.2%

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

   3. Taylor expanded in c around inf 69.6%

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

   if -5.2000000000000001e74 < c < 4600

   1. Initial program 69.1%

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

   3. Taylor expanded in c around 0 68.2%

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

      1. associate-*r/ 68.2%

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

      2. neg-mul-1 68.2%

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

   5. Simplified 68.2%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.2 \cdot 10^{+74} \lor \neg \left(c \leq 4600\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.3%

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

3. Taylor expanded in c around inf 41.6%

   \[\leadsto \color{blue}{\frac{b}{c}} \]
4. Add Preprocessing

Developer target: 99.2% 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 2024085 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (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))))