Complex division, imag part

Percentage Accurate: 61.4% → 96.0%

Time: 9.4s

Alternatives: 11

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: 61.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.0% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (a b c d)
 :precision binary64
 (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) {
	return fma((c / hypot(c, d)), (b / hypot(c, d)), (a * ((d / hypot(d, c)) / -hypot(d, c))));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

   1. div-sub 63.3%

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

   2. *-commutative 63.3%

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

   3. add-sqr-sqrt 63.3%

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

   4. times-frac 65.3%

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

   5. fma-neg 65.3%

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

   6. hypot-define 65.3%

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

   7. hypot-define 78.1%

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

   8. associate-/l* 79.7%

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

   9. add-sqr-sqrt 79.7%

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

   10. pow2 79.7%

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

   11. hypot-define 79.7%

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

4. Applied egg-rr 79.7%

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

   1. *-un-lft-identity 79.7%

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

   2. unpow2 79.7%

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

   3. times-frac 95.5%

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

6. Applied egg-rr 95.5%

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

   1. associate-*l/ 95.5%

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

   2. *-lft-identity 95.5%

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

   3. hypot-undefine 79.7%

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

   4. unpow2 79.7%

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

   5. unpow2 79.7%

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

   6. +-commutative 79.7%

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

   7. unpow2 79.7%

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

   8. unpow2 79.7%

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

   9. hypot-define 95.5%

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

   10. hypot-undefine 79.7%

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

   11. unpow2 79.7%

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

   12. unpow2 79.7%

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

   13. +-commutative 79.7%

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

   14. unpow2 79.7%

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

   15. unpow2 79.7%

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

   16. hypot-define 95.5%

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

8. Simplified 95.5%

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

9. Final simplification 95.5%

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

Alternative 2: 89.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;d \leq -2 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_1, a \cdot \frac{--1}{\mathsf{hypot}\left(d, c\right)}\right)\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_1, a \cdot \frac{-d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ c (hypot c d))) (t_1 (/ b (hypot c d))))
   (if (<= d -2e+17)
     (fma t_0 t_1 (* a (/ (- -1.0) (hypot d c))))
     (if (<= d 1.35e+154)
       (fma t_0 t_1 (* a (/ (- d) (pow (hypot c d) 2.0))))
       (/ (- (/ c (/ d b)) a) d)))))

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 (d <= -2e+17) {
		tmp = fma(t_0, t_1, (a * (-(-1.0) / hypot(d, c))));
	} else if (d <= 1.35e+154) {
		tmp = fma(t_0, t_1, (a * (-d / pow(hypot(c, d), 2.0))));
	} else {
		tmp = ((c / (d / b)) - a) / d;
	}
	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 (d <= -2e+17)
		tmp = fma(t_0, t_1, Float64(a * Float64(Float64(-(-1.0)) / hypot(d, c))));
	elseif (d <= 1.35e+154)
		tmp = fma(t_0, t_1, Float64(a * Float64(Float64(-d) / (hypot(c, d) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(c / Float64(d / b)) - a) / d);
	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[d, -2e+17], N[(t$95$0 * t$95$1 + N[(a * N[((--1.0) / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.35e+154], N[(t$95$0 * t$95$1 + N[(a * N[((-d) / N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / N[(d / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2e17

   1. Initial program 44.7%

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

      1. div-sub 44.7%

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

      2. *-commutative 44.7%

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

      3. add-sqr-sqrt 44.7%

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

      4. times-frac 45.1%

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

      5. fma-neg 45.1%

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

      6. hypot-define 45.1%

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

      7. hypot-define 59.1%

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

      8. associate-/l* 61.9%

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

      9. add-sqr-sqrt 61.9%

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

      10. pow2 61.9%

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

      11. hypot-define 61.9%

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

   4. Applied egg-rr 61.9%

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

      1. *-un-lft-identity 61.9%

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

      2. unpow2 61.9%

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

      3. times-frac 91.4%

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

   6. Applied egg-rr 91.4%

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

      1. associate-*l/ 91.4%

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

      2. *-lft-identity 91.4%

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

      3. hypot-undefine 62.0%

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

      4. unpow2 62.0%

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

      5. unpow2 62.0%

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

      6. +-commutative 62.0%

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

      7. unpow2 62.0%

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

      8. unpow2 62.0%

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

      9. hypot-define 91.4%

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

      10. hypot-undefine 62.0%

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

      11. unpow2 62.0%

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

      12. unpow2 62.0%

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

      13. +-commutative 62.0%

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

      14. unpow2 62.0%

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

      15. unpow2 62.0%

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

      16. hypot-define 91.4%

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

   8. Simplified 91.4%

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

   9. Taylor expanded in d around -inf 85.1%

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

   if -2e17 < d < 1.35000000000000003e154

   1. Initial program 77.0%

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

      1. div-sub 74.1%

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

      2. *-commutative 74.1%

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

      3. add-sqr-sqrt 74.1%

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

      4. times-frac 76.9%

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

      5. fma-neg 76.9%

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

      6. hypot-define 76.9%

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

      7. hypot-define 90.4%

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

      8. associate-/l* 91.6%

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

      9. add-sqr-sqrt 91.6%

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

      10. pow2 91.6%

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

      11. hypot-define 91.6%

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

   4. Applied egg-rr 91.6%

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

   if 1.35000000000000003e154 < d

   1. Initial program 31.3%

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

   3. Taylor expanded in c around 0 78.3%

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

      1. +-commutative 78.3%

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

      2. mul-1-neg 78.3%

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

      3. unsub-neg 78.3%

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

      4. unpow2 78.3%

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

      5. associate-/r* 87.3%

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

      6. div-sub 87.3%

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

      7. *-commutative 87.3%

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

      8. associate-/l* 90.5%

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

   5. Simplified 90.5%

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

      1. clear-num 90.5%

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

      2. un-div-inv 90.6%

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

   7. Applied egg-rr 90.6%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{--1}{\mathsf{hypot}\left(d, c\right)}\right)\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{-d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.85 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{--1}{\mathsf{hypot}\left(d, c\right)}\right)\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-62}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.85e-25)
   (fma (/ c (hypot c d)) (/ b (hypot c d)) (* a (/ (- -1.0) (hypot d c))))
   (if (<= d 1.75e-62)
     (/ (- b (* d (/ a c))) c)
     (if (<= d 5.5e+47)
       (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))
       (/ (- (/ c (/ d b)) a) d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.85e-25) {
		tmp = fma((c / hypot(c, d)), (b / hypot(c, d)), (a * (-(-1.0) / hypot(d, c))));
	} else if (d <= 1.75e-62) {
		tmp = (b - (d * (a / c))) / c;
	} else if (d <= 5.5e+47) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else {
		tmp = ((c / (d / b)) - a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.85e-25)
		tmp = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(a * Float64(Float64(-(-1.0)) / hypot(d, c))));
	elseif (d <= 1.75e-62)
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	elseif (d <= 5.5e+47)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(Float64(Float64(c / Float64(d / b)) - a) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.85e-25], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[(a * N[((--1.0) / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.75e-62], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 5.5e+47], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / N[(d / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.85000000000000004e-25

   1. Initial program 50.5%

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

      1. div-sub 50.5%

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

      2. *-commutative 50.5%

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

      3. add-sqr-sqrt 50.5%

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

      4. times-frac 51.0%

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

      5. fma-neg 51.0%

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

      6. hypot-define 51.0%

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

      7. hypot-define 65.7%

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

      8. associate-/l* 68.1%

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

      9. add-sqr-sqrt 68.1%

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

      10. pow2 68.1%

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

      11. hypot-define 68.1%

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

   4. Applied egg-rr 68.1%

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

      1. *-un-lft-identity 68.1%

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

      2. unpow2 68.1%

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

      3. times-frac 92.7%

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

   6. Applied egg-rr 92.7%

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

      1. associate-*l/ 92.8%

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

      2. *-lft-identity 92.8%

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

      3. hypot-undefine 68.2%

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

      4. unpow2 68.2%

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

      5. unpow2 68.2%

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

      6. +-commutative 68.2%

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

      7. unpow2 68.2%

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

      8. unpow2 68.2%

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

      9. hypot-define 92.8%

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

      10. hypot-undefine 68.2%

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

      11. unpow2 68.2%

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

      12. unpow2 68.2%

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

      13. +-commutative 68.2%

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

      14. unpow2 68.2%

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

      15. unpow2 68.2%

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

      16. hypot-define 92.8%

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

   8. Simplified 92.8%

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

   9. Taylor expanded in d around -inf 84.4%

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

   if -1.85000000000000004e-25 < d < 1.7500000000000001e-62

   1. Initial program 75.5%

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

   3. Taylor expanded in c around inf 90.5%

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

      1. mul-1-neg 90.5%

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

      2. unsub-neg 90.5%

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

      3. *-commutative 90.5%

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

      4. associate-/l* 91.3%

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

   5. Simplified 91.3%

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

   if 1.7500000000000001e-62 < d < 5.4999999999999998e47

   1. Initial program 95.2%

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

   if 5.4999999999999998e47 < d

   1. Initial program 42.2%

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

   3. Taylor expanded in c around 0 74.9%

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

      1. +-commutative 74.9%

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

      2. mul-1-neg 74.9%

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

      3. unsub-neg 74.9%

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

      4. unpow2 74.9%

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

      5. associate-/r* 80.4%

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

      6. div-sub 80.4%

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

      7. *-commutative 80.4%

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

      8. associate-/l* 84.3%

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

   5. Simplified 84.3%

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

      1. clear-num 84.3%

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

      2. un-div-inv 84.4%

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

   7. Applied egg-rr 84.4%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.85 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{--1}{\mathsf{hypot}\left(d, c\right)}\right)\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-62}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -6.2 \cdot 10^{+35}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d}}{d} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -5.1 \cdot 10^{-117}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-61}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{+47}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (<= d -6.2e+35)
     (- (/ (* c (/ b d)) d) (/ a d))
     (if (<= d -5.1e-117)
       t_0
       (if (<= d 2.2e-61)
         (/ (- b (* d (/ a c))) c)
         (if (<= d 1.45e+47) t_0 (/ (- (/ c (/ d b)) a) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -6.2e+35) {
		tmp = ((c * (b / d)) / d) - (a / d);
	} else if (d <= -5.1e-117) {
		tmp = t_0;
	} else if (d <= 2.2e-61) {
		tmp = (b - (d * (a / c))) / c;
	} else if (d <= 1.45e+47) {
		tmp = t_0;
	} else {
		tmp = ((c / (d / b)) - 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 = ((c * b) - (d * a)) / ((c * c) + (d * d))
    if (d <= (-6.2d+35)) then
        tmp = ((c * (b / d)) / d) - (a / d)
    else if (d <= (-5.1d-117)) then
        tmp = t_0
    else if (d <= 2.2d-61) then
        tmp = (b - (d * (a / c))) / c
    else if (d <= 1.45d+47) then
        tmp = t_0
    else
        tmp = ((c / (d / b)) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -6.2e+35) {
		tmp = ((c * (b / d)) / d) - (a / d);
	} else if (d <= -5.1e-117) {
		tmp = t_0;
	} else if (d <= 2.2e-61) {
		tmp = (b - (d * (a / c))) / c;
	} else if (d <= 1.45e+47) {
		tmp = t_0;
	} else {
		tmp = ((c / (d / b)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -6.2e+35:
		tmp = ((c * (b / d)) / d) - (a / d)
	elif d <= -5.1e-117:
		tmp = t_0
	elif d <= 2.2e-61:
		tmp = (b - (d * (a / c))) / c
	elif d <= 1.45e+47:
		tmp = t_0
	else:
		tmp = ((c / (d / b)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -6.2e+35)
		tmp = Float64(Float64(Float64(c * Float64(b / d)) / d) - Float64(a / d));
	elseif (d <= -5.1e-117)
		tmp = t_0;
	elseif (d <= 2.2e-61)
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	elseif (d <= 1.45e+47)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(c / Float64(d / b)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -6.2e+35)
		tmp = ((c * (b / d)) / d) - (a / d);
	elseif (d <= -5.1e-117)
		tmp = t_0;
	elseif (d <= 2.2e-61)
		tmp = (b - (d * (a / c))) / c;
	elseif (d <= 1.45e+47)
		tmp = t_0;
	else
		tmp = ((c / (d / b)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.2e+35], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5.1e-117], t$95$0, If[LessEqual[d, 2.2e-61], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.45e+47], t$95$0, N[(N[(N[(c / N[(d / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -6.19999999999999973e35

   1. Initial program 38.6%

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

   3. Taylor expanded in c around 0 64.7%

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

      1. +-commutative 64.7%

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

      2. mul-1-neg 64.7%

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

      3. unsub-neg 64.7%

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

      4. unpow2 64.7%

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

      5. associate-/r* 68.0%

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

      6. div-sub 68.0%

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

      7. *-commutative 68.0%

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

      8. associate-/l* 82.7%

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

   5. Simplified 82.7%

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

      1. div-sub 82.8%

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

   7. Applied egg-rr 82.8%

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

   if -6.19999999999999973e35 < d < -5.1000000000000002e-117 or 2.20000000000000009e-61 < d < 1.4499999999999999e47

   1. Initial program 90.6%

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

   if -5.1000000000000002e-117 < d < 2.20000000000000009e-61

   1. Initial program 73.9%

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

   3. Taylor expanded in c around inf 93.6%

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

      1. mul-1-neg 93.6%

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

      2. unsub-neg 93.6%

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

      3. *-commutative 93.6%

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

      4. associate-/l* 94.5%

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

   5. Simplified 94.5%

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

   if 1.4499999999999999e47 < d

   1. Initial program 42.2%

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

   3. Taylor expanded in c around 0 74.9%

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

      1. +-commutative 74.9%

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

      2. mul-1-neg 74.9%

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

      3. unsub-neg 74.9%

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

      4. unpow2 74.9%

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

      5. associate-/r* 80.4%

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

      6. div-sub 80.4%

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

      7. *-commutative 80.4%

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

      8. associate-/l* 84.3%

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

   5. Simplified 84.3%

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

      1. clear-num 84.3%

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

      2. un-div-inv 84.4%

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

   7. Applied egg-rr 84.4%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.2 \cdot 10^{+35}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d}}{d} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -5.1 \cdot 10^{-117}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-61}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{+47}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.35 \cdot 10^{-14} \lor \neg \left(d \leq 3.1 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.35e-14) (not (<= d 3.1e-52)))
   (/ (- (* c (/ b d)) a) d)
   (/ (- b (* a (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.35e-14) || !(d <= 3.1e-52)) {
		tmp = ((c * (b / d)) - 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 <= (-2.35d-14)) .or. (.not. (d <= 3.1d-52))) then
        tmp = ((c * (b / d)) - 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 <= -2.35e-14) || !(d <= 3.1e-52)) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2.35e-14) or not (d <= 3.1e-52):
		tmp = ((c * (b / d)) - a) / d
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.35e-14) || !(d <= 3.1e-52))
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - 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 <= -2.35e-14) || ~((d <= 3.1e-52)))
		tmp = ((c * (b / d)) - a) / d;
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.35e-14], N[Not[LessEqual[d, 3.1e-52]], $MachinePrecision]], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / 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 < -2.3500000000000001e-14 or 3.0999999999999999e-52 < d

   1. Initial program 55.6%

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

   3. Taylor expanded in c around 0 68.5%

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

      1. +-commutative 68.5%

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

      2. mul-1-neg 68.5%

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

      3. unsub-neg 68.5%

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

      4. unpow2 68.5%

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

      5. associate-/r* 71.7%

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

      6. div-sub 71.8%

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

      7. *-commutative 71.8%

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

      8. associate-/l* 76.9%

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

   5. Simplified 76.9%

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

   if -2.3500000000000001e-14 < d < 3.0999999999999999e-52

   1. Initial program 75.2%

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

      1. div-sub 71.2%

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

      2. *-commutative 71.2%

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

      3. add-sqr-sqrt 71.2%

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

      4. times-frac 76.4%

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

      5. fma-neg 76.4%

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

      6. hypot-define 76.4%

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

      7. hypot-define 92.2%

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

      8. associate-/l* 90.8%

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

      9. add-sqr-sqrt 90.8%

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

      10. pow2 90.8%

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

      11. hypot-define 90.8%

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

   4. Applied egg-rr 90.8%

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

   5. Taylor expanded in c around inf 88.8%

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

      1. mul-1-neg 88.8%

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

      2. unsub-neg 88.8%

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

      3. associate-/l* 89.5%

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

   7. Simplified 89.5%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.35 \cdot 10^{-14} \lor \neg \left(d \leq 3.1 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.1 \cdot 10^{+32} \lor \neg \left(d \leq 8200000000\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -3.1e+32) (not (<= d 8200000000.0)))
   (/ a (- d))
   (/ (- b (* a (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3.1e+32) || !(d <= 8200000000.0)) {
		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 <= (-3.1d+32)) .or. (.not. (d <= 8200000000.0d0))) 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 <= -3.1e+32) || !(d <= 8200000000.0)) {
		tmp = a / -d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -3.1e+32) or not (d <= 8200000000.0):
		tmp = a / -d
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -3.1e+32) || !(d <= 8200000000.0))
		tmp = Float64(a / Float64(-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 <= -3.1e+32) || ~((d <= 8200000000.0)))
		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, -3.1e+32], N[Not[LessEqual[d, 8200000000.0]], $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 < -3.09999999999999993e32 or 8.2e9 < d

   1. Initial program 45.5%

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

   3. Taylor expanded in c around 0 67.8%

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

      1. associate-*r/ 67.8%

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

      2. neg-mul-1 67.8%

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

   5. Simplified 67.8%

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

   if -3.09999999999999993e32 < d < 8.2e9

   1. Initial program 78.7%

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

      1. div-sub 75.3%

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

      2. *-commutative 75.3%

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

      3. add-sqr-sqrt 75.3%

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

      4. times-frac 78.5%

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

      5. fma-neg 78.5%

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

      6. hypot-define 78.5%

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

      7. hypot-define 92.2%

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

      8. associate-/l* 91.1%

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

      9. add-sqr-sqrt 91.1%

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

      10. pow2 91.1%

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

      11. hypot-define 91.1%

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

   4. Applied egg-rr 91.1%

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

   5. Taylor expanded in c around inf 81.1%

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

      1. mul-1-neg 81.1%

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

      2. unsub-neg 81.1%

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

      3. associate-/l* 81.7%

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

   7. Simplified 81.7%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.1 \cdot 10^{+32} \lor \neg \left(d \leq 8200000000\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{-14}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d}}{d} - \frac{a}{d}\\ \mathbf{elif}\;d \leq 3.15 \cdot 10^{-52}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.15e-14)
   (- (/ (* c (/ b d)) d) (/ a d))
   (if (<= d 3.15e-52) (/ (- b (* a (/ d c))) c) (/ (- (/ c (/ d b)) a) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.15e-14) {
		tmp = ((c * (b / d)) / d) - (a / d);
	} else if (d <= 3.15e-52) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((c / (d / b)) - 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 (d <= (-1.15d-14)) then
        tmp = ((c * (b / d)) / d) - (a / d)
    else if (d <= 3.15d-52) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = ((c / (d / b)) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.15e-14) {
		tmp = ((c * (b / d)) / d) - (a / d);
	} else if (d <= 3.15e-52) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((c / (d / b)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.15e-14:
		tmp = ((c * (b / d)) / d) - (a / d)
	elif d <= 3.15e-52:
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = ((c / (d / b)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.15e-14)
		tmp = Float64(Float64(Float64(c * Float64(b / d)) / d) - Float64(a / d));
	elseif (d <= 3.15e-52)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(Float64(c / Float64(d / b)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -1.15e-14)
		tmp = ((c * (b / d)) / d) - (a / d);
	elseif (d <= 3.15e-52)
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = ((c / (d / b)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.15e-14], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.15e-52], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(c / N[(d / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.14999999999999999e-14

   1. Initial program 48.7%

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

   3. Taylor expanded in c around 0 63.6%

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

      1. +-commutative 63.6%

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

      2. mul-1-neg 63.6%

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

      3. unsub-neg 63.6%

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

      4. unpow2 63.6%

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

      5. associate-/r* 66.4%

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

      6. div-sub 66.4%

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

      7. *-commutative 66.4%

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

      8. associate-/l* 78.7%

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

   5. Simplified 78.7%

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

      1. div-sub 78.7%

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

   7. Applied egg-rr 78.7%

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

   if -1.14999999999999999e-14 < d < 3.1500000000000002e-52

   1. Initial program 75.2%

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

      1. div-sub 71.2%

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

      2. *-commutative 71.2%

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

      3. add-sqr-sqrt 71.2%

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

      4. times-frac 76.4%

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

      5. fma-neg 76.4%

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

      6. hypot-define 76.4%

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

      7. hypot-define 92.2%

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

      8. associate-/l* 90.8%

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

      9. add-sqr-sqrt 90.8%

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

      10. pow2 90.8%

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

      11. hypot-define 90.8%

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

   4. Applied egg-rr 90.8%

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

   5. Taylor expanded in c around inf 88.8%

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

      1. mul-1-neg 88.8%

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

      2. unsub-neg 88.8%

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

      3. associate-/l* 89.5%

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

   7. Simplified 89.5%

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

   if 3.1500000000000002e-52 < d

   1. Initial program 60.5%

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

   3. Taylor expanded in c around 0 72.0%

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

      1. +-commutative 72.0%

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

      2. mul-1-neg 72.0%

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

      3. unsub-neg 72.0%

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

      4. unpow2 72.0%

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

      5. associate-/r* 75.6%

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

      6. div-sub 75.6%

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

      7. *-commutative 75.6%

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

      8. associate-/l* 75.6%

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

   5. Simplified 75.6%

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

      1. clear-num 75.6%

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

      2. un-div-inv 75.7%

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

   7. Applied egg-rr 75.7%

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

Alternative 8: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-52}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.1e-14)
   (/ (- (* c (/ b d)) a) d)
   (if (<= d 4e-52) (/ (- b (* a (/ d c))) c) (/ (- (/ c (/ d b)) a) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.1e-14) {
		tmp = ((c * (b / d)) - a) / d;
	} else if (d <= 4e-52) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((c / (d / b)) - 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 (d <= (-1.1d-14)) then
        tmp = ((c * (b / d)) - a) / d
    else if (d <= 4d-52) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = ((c / (d / b)) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.1e-14) {
		tmp = ((c * (b / d)) - a) / d;
	} else if (d <= 4e-52) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((c / (d / b)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.1e-14:
		tmp = ((c * (b / d)) - a) / d
	elif d <= 4e-52:
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = ((c / (d / b)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.1e-14)
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	elseif (d <= 4e-52)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(Float64(c / Float64(d / b)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -1.1e-14)
		tmp = ((c * (b / d)) - a) / d;
	elseif (d <= 4e-52)
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = ((c / (d / b)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.1e-14], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 4e-52], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(c / N[(d / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.1e-14

   1. Initial program 48.7%

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

   3. Taylor expanded in c around 0 63.6%

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

      1. +-commutative 63.6%

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

      2. mul-1-neg 63.6%

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

      3. unsub-neg 63.6%

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

      4. unpow2 63.6%

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

      5. associate-/r* 66.4%

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

      6. div-sub 66.4%

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

      7. *-commutative 66.4%

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

      8. associate-/l* 78.7%

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

   5. Simplified 78.7%

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

   if -1.1e-14 < d < 4e-52

   1. Initial program 75.2%

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

      1. div-sub 71.2%

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

      2. *-commutative 71.2%

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

      3. add-sqr-sqrt 71.2%

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

      4. times-frac 76.4%

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

      5. fma-neg 76.4%

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

      6. hypot-define 76.4%

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

      7. hypot-define 92.2%

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

      8. associate-/l* 90.8%

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

      9. add-sqr-sqrt 90.8%

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

      10. pow2 90.8%

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

      11. hypot-define 90.8%

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

   4. Applied egg-rr 90.8%

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

   5. Taylor expanded in c around inf 88.8%

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

      1. mul-1-neg 88.8%

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

      2. unsub-neg 88.8%

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

      3. associate-/l* 89.5%

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

   7. Simplified 89.5%

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

   if 4e-52 < d

   1. Initial program 60.5%

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

   3. Taylor expanded in c around 0 72.0%

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

      1. +-commutative 72.0%

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

      2. mul-1-neg 72.0%

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

      3. unsub-neg 72.0%

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

      4. unpow2 72.0%

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

      5. associate-/r* 75.6%

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

      6. div-sub 75.6%

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

      7. *-commutative 75.6%

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

      8. associate-/l* 75.6%

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

   5. Simplified 75.6%

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

      1. clear-num 75.6%

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

      2. un-div-inv 75.7%

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

   7. Applied egg-rr 75.7%

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

Alternative 9: 62.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -9.5 \cdot 10^{-107} \lor \neg \left(c \leq 9500000000000\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -9.5e-107) (not (<= c 9500000000000.0))) (/ b c) (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -9.5e-107) || !(c <= 9500000000000.0)) {
		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 <= (-9.5d-107)) .or. (.not. (c <= 9500000000000.0d0))) 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 <= -9.5e-107) || !(c <= 9500000000000.0)) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -9.5e-107) or not (c <= 9500000000000.0):
		tmp = b / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -9.5e-107) || !(c <= 9500000000000.0))
		tmp = Float64(b / c);
	else
		tmp = Float64(a / Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -9.5e-107) || ~((c <= 9500000000000.0)))
		tmp = b / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -9.5e-107], N[Not[LessEqual[c, 9500000000000.0]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -9.4999999999999999e-107 or 9.5e12 < c

   1. Initial program 58.4%

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

   3. Taylor expanded in c around inf 61.3%

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

   if -9.4999999999999999e-107 < c < 9.5e12

   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 0 65.0%

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

      1. associate-*r/ 65.0%

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

      2. neg-mul-1 65.0%

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

   5. Simplified 65.0%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9.5 \cdot 10^{-107} \lor \neg \left(c \leq 9500000000000\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+225} \lor \neg \left(d \leq 4.9 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.9e+225) (not (<= d 4.9e+154))) (/ a d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.9e+225) || !(d <= 4.9e+154)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-1.9d+225)) .or. (.not. (d <= 4.9d+154))) then
        tmp = a / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.9e+225) || !(d <= 4.9e+154)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.9e+225) or not (d <= 4.9e+154):
		tmp = a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.9e+225) || !(d <= 4.9e+154))
		tmp = Float64(a / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.9e+225) || ~((d <= 4.9e+154)))
		tmp = a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.9e+225], N[Not[LessEqual[d, 4.9e+154]], $MachinePrecision]], N[(a / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.9e225 or 4.9000000000000002e154 < d

   1. Initial program 33.4%

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

   3. Taylor expanded in c around 0 84.2%

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

      1. associate-*r/ 84.2%

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

      2. neg-mul-1 84.2%

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

   5. Simplified 84.2%

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

      1. neg-sub0 84.2%

         \[\leadsto \frac{\color{blue}{0 - a}}{d} \]

      2. sub-neg 84.2%

         \[\leadsto \frac{\color{blue}{0 + \left(-a\right)}}{d} \]

      3. add-sqr-sqrt 44.0%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{-a} \cdot \sqrt{-a}}}{d} \]

      4. sqrt-unprod 43.9%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}}{d} \]

      5. sqr-neg 43.9%

         \[\leadsto \frac{0 + \sqrt{\color{blue}{a \cdot a}}}{d} \]

      6. sqrt-unprod 17.9%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{a} \cdot \sqrt{a}}}{d} \]

      7. add-sqr-sqrt 32.3%

         \[\leadsto \frac{0 + \color{blue}{a}}{d} \]

   7. Applied egg-rr 32.3%

      \[\leadsto \frac{\color{blue}{0 + a}}{d} \]
   8. Step-by-step derivation

      1. +-lft-identity 32.3%

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

   9. Simplified 32.3%

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

   if -1.9e225 < d < 4.9000000000000002e154

   1. Initial program 71.8%

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

   3. Taylor expanded in c around inf 47.1%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+225} \lor \neg \left(d \leq 4.9 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 11.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

3. Taylor expanded in c around 0 39.9%

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

   1. associate-*r/ 39.9%

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

   2. neg-mul-1 39.9%

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

5. Simplified 39.9%

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

   1. neg-sub0 39.9%

      \[\leadsto \frac{\color{blue}{0 - a}}{d} \]

   2. sub-neg 39.9%

      \[\leadsto \frac{\color{blue}{0 + \left(-a\right)}}{d} \]

   3. add-sqr-sqrt 18.7%

      \[\leadsto \frac{0 + \color{blue}{\sqrt{-a} \cdot \sqrt{-a}}}{d} \]

   4. sqrt-unprod 18.6%

      \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}}{d} \]

   5. sqr-neg 18.6%

      \[\leadsto \frac{0 + \sqrt{\color{blue}{a \cdot a}}}{d} \]

   6. sqrt-unprod 4.7%

      \[\leadsto \frac{0 + \color{blue}{\sqrt{a} \cdot \sqrt{a}}}{d} \]

   7. add-sqr-sqrt 9.5%

      \[\leadsto \frac{0 + \color{blue}{a}}{d} \]

7. Applied egg-rr 9.5%

   \[\leadsto \frac{\color{blue}{0 + a}}{d} \]
8. Step-by-step derivation

   1. +-lft-identity 9.5%

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

9. Simplified 9.5%

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

Developer Target 1: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024144 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (! :herbie-platform default (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))))