Complex division, imag part

Percentage Accurate: 61.7% → 97.6%

Time: 14.3s

Alternatives: 14

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.7% 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: 97.6% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (a b c d)
 :precision binary64
 (fma
  (/ c (hypot c d))
  (/ b (hypot c d))
  (/ (* d (- (/ a (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)), ((d * -(a / 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(Float64(d * Float64(-Float64(a / hypot(d, c)))) / 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[(N[(d * (-N[(a / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.8%

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

   1. div-sub 56.4%

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

   2. sub-neg 56.4%

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

   3. *-commutative 56.4%

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

   4. add-sqr-sqrt 56.4%

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

   5. times-frac 59.1%

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

   6. fma-def 59.1%

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

   7. hypot-def 59.1%

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

   8. hypot-def 69.1%

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

   9. associate-/l* 73.9%

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

   10. add-sqr-sqrt 73.9%

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

   11. pow2 73.9%

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

   12. hypot-def 73.9%

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

3. Applied egg-rr 73.9%

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

   1. unpow2 73.9%

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

   2. *-un-lft-identity 73.9%

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

   3. times-frac 94.4%

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

5. Applied egg-rr 94.4%

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

   1. /-rgt-identity 94.4%

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

   2. *-un-lft-identity 94.4%

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

   3. times-frac 96.8%

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

7. Applied egg-rr 96.8%

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

   1. associate-*l/ 97.0%

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

   2. *-lft-identity 97.0%

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

   3. associate-/r/ 97.7%

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

   4. hypot-def 75.0%

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

   5. unpow2 75.0%

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

   6. unpow2 75.0%

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

   7. +-commutative 75.0%

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

   8. unpow2 75.0%

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

   9. unpow2 75.0%

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

   10. hypot-def 97.7%

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

   11. hypot-def 75.0%

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

   12. unpow2 75.0%

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

   13. unpow2 75.0%

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

   14. +-commutative 75.0%

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

   15. unpow2 75.0%

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

   16. unpow2 75.0%

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

   17. hypot-def 97.7%

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

9. Simplified 97.7%

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

10. Final simplification 97.7%

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

Alternative 2: 89.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ t_2 := \mathsf{fma}\left(t_0, t_1, \frac{-a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)\\ t_3 := c \cdot b - d \cdot a\\ t_4 := \frac{t_3}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_4 \leq -4 \cdot 10^{+291}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_4 \leq 2 \cdot 10^{+142}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_3}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_1, \frac{-a}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ c (hypot c d)))
        (t_1 (/ b (hypot c d)))
        (t_2 (fma t_0 t_1 (/ (- a) (/ (pow (hypot c d) 2.0) d))))
        (t_3 (- (* c b) (* d a)))
        (t_4 (/ t_3 (+ (* c c) (* d d)))))
   (if (<= t_4 -4e+291)
     t_2
     (if (<= t_4 2e+142)
       (* (/ 1.0 (hypot c d)) (/ t_3 (hypot c d)))
       (if (<= t_4 INFINITY) t_2 (fma t_0 t_1 (/ (- 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 t_2 = fma(t_0, t_1, (-a / (pow(hypot(c, d), 2.0) / d)));
	double t_3 = (c * b) - (d * a);
	double t_4 = t_3 / ((c * c) + (d * d));
	double tmp;
	if (t_4 <= -4e+291) {
		tmp = t_2;
	} else if (t_4 <= 2e+142) {
		tmp = (1.0 / hypot(c, d)) * (t_3 / hypot(c, d));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = fma(t_0, t_1, (-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))
	t_2 = fma(t_0, t_1, Float64(Float64(-a) / Float64((hypot(c, d) ^ 2.0) / d)))
	t_3 = Float64(Float64(c * b) - Float64(d * a))
	t_4 = Float64(t_3 / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (t_4 <= -4e+291)
		tmp = t_2;
	elseif (t_4 <= 2e+142)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_3 / hypot(c, d)));
	elseif (t_4 <= Inf)
		tmp = t_2;
	else
		tmp = fma(t_0, t_1, Float64(Float64(-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]}, Block[{t$95$2 = N[(t$95$0 * t$95$1 + N[((-a) / N[(N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -4e+291], t$95$2, If[LessEqual[t$95$4, 2e+142], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$3 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$2, N[(t$95$0 * t$95$1 + N[((-a) / d), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < -3.9999999999999998e291 or 2.0000000000000001e142 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < +inf.0

   1. Initial program 58.3%

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

      1. div-sub 45.5%

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

      2. sub-neg 45.5%

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

      3. *-commutative 45.5%

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

      4. add-sqr-sqrt 45.5%

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

      5. times-frac 67.7%

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

      6. fma-def 67.7%

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

      7. hypot-def 67.7%

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

      8. hypot-def 69.9%

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

      9. associate-/l* 94.0%

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

      10. add-sqr-sqrt 94.0%

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

      11. pow2 94.0%

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

      12. hypot-def 94.0%

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

   3. Applied egg-rr 94.0%

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

   if -3.9999999999999998e291 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 2.0000000000000001e142

   1. Initial program 79.5%

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

      1. *-un-lft-identity 79.5%

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

      2. add-sqr-sqrt 79.5%

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

      3. times-frac 79.6%

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

      4. hypot-def 79.6%

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

      5. hypot-def 99.6%

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

   3. Applied egg-rr 99.6%

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

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

   1. Initial program 0.0%

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

      1. div-sub 0.0%

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

      2. sub-neg 0.0%

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

      3. *-commutative 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. times-frac 1.0%

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

      6. fma-def 1.0%

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

      7. hypot-def 1.0%

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

      8. hypot-def 27.7%

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

      9. associate-/l* 34.0%

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

      10. add-sqr-sqrt 34.0%

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

      11. pow2 34.0%

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

      12. hypot-def 34.0%

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

   3. Applied egg-rr 34.0%

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

   4. Taylor expanded in c around 0 75.6%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq -4 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)\\ \mathbf{elif}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+142}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \end{array} \]

Alternative 3: 89.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := c \cdot b - d \cdot a\\ t_2 := \frac{t_1}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_0 \cdot \frac{c}{\frac{\mathsf{hypot}\left(c, d\right)}{b}} - \frac{d}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{a}}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+252}:\\ \;\;\;\;t_0 \cdot \frac{t_1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot c d)))
        (t_1 (- (* c b) (* d a)))
        (t_2 (/ t_1 (+ (* c c) (* d d)))))
   (if (<= t_2 (- INFINITY))
     (- (* t_0 (/ c (/ (hypot c d) b))) (/ d (/ (pow (hypot c d) 2.0) a)))
     (if (<= t_2 2e+252)
       (* t_0 (/ t_1 (hypot c d)))
       (fma (/ c (hypot c d)) (/ b (hypot c d)) (/ (- a) d))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = 1.0 / hypot(c, d);
	double t_1 = (c * b) - (d * a);
	double t_2 = t_1 / ((c * c) + (d * d));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (t_0 * (c / (hypot(c, d) / b))) - (d / (pow(hypot(c, d), 2.0) / a));
	} else if (t_2 <= 2e+252) {
		tmp = t_0 * (t_1 / hypot(c, d));
	} else {
		tmp = fma((c / hypot(c, d)), (b / hypot(c, d)), (-a / d));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(1.0 / hypot(c, d))
	t_1 = Float64(Float64(c * b) - Float64(d * a))
	t_2 = Float64(t_1 / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 * Float64(c / Float64(hypot(c, d) / b))) - Float64(d / Float64((hypot(c, d) ^ 2.0) / a)));
	elseif (t_2 <= 2e+252)
		tmp = Float64(t_0 * Float64(t_1 / hypot(c, d)));
	else
		tmp = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(Float64(-a) / d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(t$95$0 * N[(c / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(d / N[(N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+252], N[(t$95$0 * N[(t$95$1 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[((-a) / d), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 45.5%

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

      1. div-sub 16.5%

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

      2. sub-neg 16.5%

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

      3. *-un-lft-identity 16.5%

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

      4. add-sqr-sqrt 16.5%

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

      5. times-frac 16.5%

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

      6. fma-def 16.5%

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

      7. hypot-def 16.5%

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

      8. hypot-def 38.3%

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

      9. associate-/l* 66.0%

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

      10. add-sqr-sqrt 66.0%

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

      11. pow2 66.0%

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

      12. hypot-def 66.0%

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

   3. Applied egg-rr 66.0%

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

      1. fma-neg 58.9%

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

      2. *-commutative 58.9%

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

      3. associate-/l* 85.9%

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

      4. associate-/l* 58.2%

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

      5. *-commutative 58.2%

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

      6. associate-/l* 85.9%

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

   5. Simplified 85.9%

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

   if -inf.0 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 2.0000000000000002e252

   1. Initial program 81.0%

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

      1. *-un-lft-identity 81.0%

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

      2. add-sqr-sqrt 81.0%

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

      3. times-frac 81.0%

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

      4. hypot-def 81.0%

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

      5. hypot-def 99.6%

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

   3. Applied egg-rr 99.6%

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

   if 2.0000000000000002e252 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 12.7%

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

      1. div-sub 10.0%

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

      2. sub-neg 10.0%

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

      3. *-commutative 10.0%

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

      4. add-sqr-sqrt 10.0%

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

      5. times-frac 18.3%

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

      6. fma-def 18.3%

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

      7. hypot-def 18.3%

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

      8. hypot-def 37.4%

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

      9. associate-/l* 51.9%

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

      10. add-sqr-sqrt 51.9%

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

      11. pow2 51.9%

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

      12. hypot-def 51.9%

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

   3. Applied egg-rr 51.9%

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

   4. Taylor expanded in c around 0 76.2%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq -\infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\frac{\mathsf{hypot}\left(c, d\right)}{b}} - \frac{d}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{a}}\\ \mathbf{elif}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+252}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \end{array} \]

Alternative 4: 88.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot b - d \cdot a\\ \mathbf{if}\;\frac{t_0}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+252}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c b) (* d a))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 2e+252)
     (* (/ 1.0 (hypot c d)) (/ t_0 (hypot c d)))
     (fma (/ c (hypot c d)) (/ b (hypot c d)) (/ (- a) d)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * b) - (d * a);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 2e+252) {
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	} else {
		tmp = fma((c / hypot(c, d)), (b / hypot(c, d)), (-a / d));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * b) - Float64(d * a))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= 2e+252)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, d)));
	else
		tmp = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(Float64(-a) / d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+252], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[((-a) / d), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 2.0000000000000002e252

   1. Initial program 78.2%

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

      1. *-un-lft-identity 78.2%

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

      2. add-sqr-sqrt 78.2%

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

      3. times-frac 78.3%

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

      4. hypot-def 78.3%

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

      5. hypot-def 95.9%

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

   3. Applied egg-rr 95.9%

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

   if 2.0000000000000002e252 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 12.7%

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

      1. div-sub 10.0%

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

      2. sub-neg 10.0%

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

      3. *-commutative 10.0%

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

      4. add-sqr-sqrt 10.0%

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

      5. times-frac 18.3%

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

      6. fma-def 18.3%

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

      7. hypot-def 18.3%

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

      8. hypot-def 37.4%

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

      9. associate-/l* 51.9%

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

      10. add-sqr-sqrt 51.9%

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

      11. pow2 51.9%

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

      12. hypot-def 51.9%

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

   3. Applied egg-rr 51.9%

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

   4. Taylor expanded in c around 0 76.2%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+252}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \end{array} \]

Alternative 5: 85.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot b - d \cdot a\\ \mathbf{if}\;\frac{t_0}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+252}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c b) (* d a))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 2e+252)
     (* (/ 1.0 (hypot c d)) (/ t_0 (hypot c d)))
     (/ (- (* c (/ b d)) a) d))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * b) - (d * a);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 2e+252) {
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (c * b) - (d * a);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 2e+252) {
		tmp = (1.0 / Math.hypot(c, d)) * (t_0 / Math.hypot(c, d));
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (c * b) - (d * a)
	tmp = 0
	if (t_0 / ((c * c) + (d * d))) <= 2e+252:
		tmp = (1.0 / math.hypot(c, d)) * (t_0 / math.hypot(c, d))
	else:
		tmp = ((c * (b / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * b) - Float64(d * a))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= 2e+252)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, d)));
	else
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (c * b) - (d * a);
	tmp = 0.0;
	if ((t_0 / ((c * c) + (d * d))) <= 2e+252)
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	else
		tmp = ((c * (b / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+252], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 2.0000000000000002e252

   1. Initial program 78.2%

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

      1. *-un-lft-identity 78.2%

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

      2. add-sqr-sqrt 78.2%

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

      3. times-frac 78.3%

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

      4. hypot-def 78.3%

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

      5. hypot-def 95.9%

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

   3. Applied egg-rr 95.9%

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

   if 2.0000000000000002e252 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 12.7%

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

   2. Taylor expanded in c around 0 59.3%

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

      1. +-commutative 59.3%

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

      2. mul-1-neg 59.3%

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

      3. unsub-neg 59.3%

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

      4. associate-/l* 61.0%

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

      5. associate-/r/ 59.9%

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

   4. Simplified 59.9%

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

      1. *-un-lft-identity 59.9%

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

      2. pow2 59.9%

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

      3. times-frac 62.5%

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

   6. Applied egg-rr 62.5%

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

      1. associate-*l/ 62.5%

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

      2. *-lft-identity 62.5%

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

   8. Simplified 62.5%

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

      1. associate-*l/ 65.0%

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

      2. sub-div 65.2%

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

   10. Applied egg-rr 65.2%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+252}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]

Alternative 6: 79.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot c + d \cdot d\\ \mathbf{if}\;d \leq -4.7 \cdot 10^{+88}:\\ \;\;\;\;c \cdot \frac{b \cdot \frac{1}{d}}{d} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, c \cdot b\right)}{t_0}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{-137}:\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{a}{{c}^{2}}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* c c) (* d d))))
   (if (<= d -4.7e+88)
     (- (* c (/ (* b (/ 1.0 d)) d)) (/ a d))
     (if (<= d -6e-95)
       (/ (fma (- d) a (* c b)) t_0)
       (if (<= d 1.8e-137)
         (- (/ b c) (* d (/ a (pow c 2.0))))
         (if (<= d 1.2e+54)
           (/ (- (* c b) (* d a)) t_0)
           (/ (- (* c (/ b d)) a) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * c) + (d * d);
	double tmp;
	if (d <= -4.7e+88) {
		tmp = (c * ((b * (1.0 / d)) / d)) - (a / d);
	} else if (d <= -6e-95) {
		tmp = fma(-d, a, (c * b)) / t_0;
	} else if (d <= 1.8e-137) {
		tmp = (b / c) - (d * (a / pow(c, 2.0)));
	} else if (d <= 1.2e+54) {
		tmp = ((c * b) - (d * a)) / t_0;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * c) + Float64(d * d))
	tmp = 0.0
	if (d <= -4.7e+88)
		tmp = Float64(Float64(c * Float64(Float64(b * Float64(1.0 / d)) / d)) - Float64(a / d));
	elseif (d <= -6e-95)
		tmp = Float64(fma(Float64(-d), a, Float64(c * b)) / t_0);
	elseif (d <= 1.8e-137)
		tmp = Float64(Float64(b / c) - Float64(d * Float64(a / (c ^ 2.0))));
	elseif (d <= 1.2e+54)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / t_0);
	else
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4.7e+88], N[(N[(c * N[(N[(b * N[(1.0 / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -6e-95], N[(N[((-d) * a + N[(c * b), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[d, 1.8e-137], N[(N[(b / c), $MachinePrecision] - N[(d * N[(a / N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.2e+54], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -4.70000000000000007e88

   1. Initial program 30.8%

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

   2. Taylor expanded in c around 0 82.4%

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

      1. +-commutative 82.4%

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

      2. mul-1-neg 82.4%

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

      3. unsub-neg 82.4%

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

      4. associate-/l* 82.5%

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

      5. associate-/r/ 80.5%

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

   4. Simplified 80.5%

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

      1. *-un-lft-identity 80.5%

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

      2. pow2 80.5%

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

      3. times-frac 82.6%

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

   6. Applied egg-rr 82.6%

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

      1. associate-*r/ 82.6%

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

   8. Simplified 82.6%

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

   if -4.70000000000000007e88 < d < -6e-95

   1. Initial program 87.4%

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

      1. sub-neg 87.4%

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

      2. +-commutative 87.4%

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

      3. *-commutative 87.4%

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

      4. distribute-lft-neg-in 87.4%

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

      5. fma-def 87.4%

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

   3. Applied egg-rr 87.4%

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

   if -6e-95 < d < 1.80000000000000003e-137

   1. Initial program 67.4%

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

   2. Taylor expanded in c around inf 85.9%

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

      1. +-commutative 85.9%

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

      2. mul-1-neg 85.9%

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

      3. unsub-neg 85.9%

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

      4. associate-/l* 84.8%

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

      5. associate-/r/ 87.4%

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

   4. Simplified 87.4%

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

   if 1.80000000000000003e-137 < d < 1.19999999999999999e54

   1. Initial program 89.0%

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

   if 1.19999999999999999e54 < d

   1. Initial program 35.0%

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

   2. Taylor expanded in c around 0 79.0%

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

      1. +-commutative 79.0%

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

      2. mul-1-neg 79.0%

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

      3. unsub-neg 79.0%

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

      4. associate-/l* 78.0%

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

      5. associate-/r/ 80.7%

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

   4. Simplified 80.7%

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

      1. *-un-lft-identity 80.7%

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

      2. pow2 80.7%

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

      3. times-frac 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. associate-*l/ 83.7%

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

      2. *-lft-identity 83.7%

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

   8. Simplified 83.7%

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

      1. associate-*l/ 87.9%

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

      2. sub-div 88.0%

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

   10. Applied egg-rr 88.0%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.7 \cdot 10^{+88}:\\ \;\;\;\;c \cdot \frac{b \cdot \frac{1}{d}}{d} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, c \cdot b\right)}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{-137}:\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{a}{{c}^{2}}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]

Alternative 7: 80.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -6.1 \cdot 10^{+88}:\\ \;\;\;\;c \cdot \frac{b \cdot \frac{1}{d}}{d} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -6.8 \cdot 10^{-96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{-130}:\\ \;\;\;\;\frac{b}{c} - a \cdot \frac{d}{{c}^{2}}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (<= d -6.1e+88)
     (- (* c (/ (* b (/ 1.0 d)) d)) (/ a d))
     (if (<= d -6.8e-96)
       t_0
       (if (<= d 1.5e-130)
         (- (/ b c) (* a (/ d (pow c 2.0))))
         (if (<= d 1.45e+54) t_0 (/ (- (* c (/ b d)) 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 <= -6.1e+88) {
		tmp = (c * ((b * (1.0 / d)) / d)) - (a / d);
	} else if (d <= -6.8e-96) {
		tmp = t_0;
	} else if (d <= 1.5e-130) {
		tmp = (b / c) - (a * (d / pow(c, 2.0)));
	} else if (d <= 1.45e+54) {
		tmp = t_0;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
    if (d <= (-6.1d+88)) then
        tmp = (c * ((b * (1.0d0 / d)) / d)) - (a / d)
    else if (d <= (-6.8d-96)) then
        tmp = t_0
    else if (d <= 1.5d-130) then
        tmp = (b / c) - (a * (d / (c ** 2.0d0)))
    else if (d <= 1.45d+54) then
        tmp = t_0
    else
        tmp = ((c * (b / d)) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -6.1e+88) {
		tmp = (c * ((b * (1.0 / d)) / d)) - (a / d);
	} else if (d <= -6.8e-96) {
		tmp = t_0;
	} else if (d <= 1.5e-130) {
		tmp = (b / c) - (a * (d / Math.pow(c, 2.0)));
	} else if (d <= 1.45e+54) {
		tmp = t_0;
	} else {
		tmp = ((c * (b / d)) - 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 <= -6.1e+88:
		tmp = (c * ((b * (1.0 / d)) / d)) - (a / d)
	elif d <= -6.8e-96:
		tmp = t_0
	elif d <= 1.5e-130:
		tmp = (b / c) - (a * (d / math.pow(c, 2.0)))
	elif d <= 1.45e+54:
		tmp = t_0
	else:
		tmp = ((c * (b / d)) - 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 <= -6.1e+88)
		tmp = Float64(Float64(c * Float64(Float64(b * Float64(1.0 / d)) / d)) - Float64(a / d));
	elseif (d <= -6.8e-96)
		tmp = t_0;
	elseif (d <= 1.5e-130)
		tmp = Float64(Float64(b / c) - Float64(a * Float64(d / (c ^ 2.0))));
	elseif (d <= 1.45e+54)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - 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 <= -6.1e+88)
		tmp = (c * ((b * (1.0 / d)) / d)) - (a / d);
	elseif (d <= -6.8e-96)
		tmp = t_0;
	elseif (d <= 1.5e-130)
		tmp = (b / c) - (a * (d / (c ^ 2.0)));
	elseif (d <= 1.45e+54)
		tmp = t_0;
	else
		tmp = ((c * (b / d)) - 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, -6.1e+88], N[(N[(c * N[(N[(b * N[(1.0 / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -6.8e-96], t$95$0, If[LessEqual[d, 1.5e-130], N[(N[(b / c), $MachinePrecision] - N[(a * N[(d / N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.45e+54], t$95$0, N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -6.0999999999999998e88

   1. Initial program 30.8%

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

   2. Taylor expanded in c around 0 82.4%

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

      1. +-commutative 82.4%

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

      2. mul-1-neg 82.4%

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

      3. unsub-neg 82.4%

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

      4. associate-/l* 82.5%

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

      5. associate-/r/ 80.5%

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

   4. Simplified 80.5%

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

      1. *-un-lft-identity 80.5%

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

      2. pow2 80.5%

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

      3. times-frac 82.6%

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

   6. Applied egg-rr 82.6%

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

      1. associate-*r/ 82.6%

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

   8. Simplified 82.6%

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

   if -6.0999999999999998e88 < d < -6.8000000000000002e-96 or 1.49999999999999993e-130 < d < 1.4499999999999999e54

   1. Initial program 88.2%

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

   if -6.8000000000000002e-96 < d < 1.49999999999999993e-130

   1. Initial program 67.4%

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

      1. div-sub 58.5%

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

      2. sub-neg 58.5%

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

      3. *-commutative 58.5%

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

      4. add-sqr-sqrt 58.5%

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

      5. times-frac 69.0%

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

      6. fma-def 69.0%

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

      7. hypot-def 69.0%

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

      8. hypot-def 88.6%

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

      9. associate-/l* 87.6%

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

      10. add-sqr-sqrt 87.6%

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

      11. pow2 87.6%

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

      12. hypot-def 87.6%

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

   3. Applied egg-rr 87.6%

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

      1. unpow2 87.6%

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

      2. *-un-lft-identity 87.6%

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

      3. times-frac 93.1%

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

   5. Applied egg-rr 93.1%

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

      1. /-rgt-identity 93.1%

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

      2. *-un-lft-identity 93.1%

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

      3. times-frac 95.7%

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

   7. Applied egg-rr 95.7%

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

      1. associate-*l/ 95.8%

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

      2. *-lft-identity 95.8%

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

      3. associate-/r/ 98.5%

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

      4. hypot-def 91.6%

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

      5. unpow2 91.6%

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

      6. unpow2 91.6%

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

      7. +-commutative 91.6%

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

      8. unpow2 91.6%

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

      9. unpow2 91.6%

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

      10. hypot-def 98.5%

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

      11. hypot-def 91.6%

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

      12. unpow2 91.6%

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

      13. unpow2 91.6%

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

      14. +-commutative 91.6%

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

      15. unpow2 91.6%

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

      16. unpow2 91.6%

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

      17. hypot-def 98.5%

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

   9. Simplified 98.5%

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

   10. Taylor expanded in c around inf 85.9%

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

       1. +-commutative 85.9%

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

       2. mul-1-neg 85.9%

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

       3. *-commutative 85.9%

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

       4. associate-*l/ 84.8%

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

       5. unsub-neg 84.8%

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

       6. *-commutative 84.8%

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

   12. Simplified 84.8%

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

   if 1.4499999999999999e54 < d

   1. Initial program 35.0%

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

   2. Taylor expanded in c around 0 79.0%

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

      1. +-commutative 79.0%

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

      2. mul-1-neg 79.0%

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

      3. unsub-neg 79.0%

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

      4. associate-/l* 78.0%

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

      5. associate-/r/ 80.7%

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

   4. Simplified 80.7%

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

      1. *-un-lft-identity 80.7%

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

      2. pow2 80.7%

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

      3. times-frac 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. associate-*l/ 83.7%

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

      2. *-lft-identity 83.7%

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

   8. Simplified 83.7%

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

      1. associate-*l/ 87.9%

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

      2. sub-div 88.0%

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

   10. Applied egg-rr 88.0%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.1 \cdot 10^{+88}:\\ \;\;\;\;c \cdot \frac{b \cdot \frac{1}{d}}{d} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -6.8 \cdot 10^{-96}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{-130}:\\ \;\;\;\;\frac{b}{c} - a \cdot \frac{d}{{c}^{2}}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{+54}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]

Alternative 8: 79.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -1.45 \cdot 10^{+89}:\\ \;\;\;\;c \cdot \frac{b \cdot \frac{1}{d}}{d} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-138}:\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{a}{{c}^{2}}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (<= d -1.45e+89)
     (- (* c (/ (* b (/ 1.0 d)) d)) (/ a d))
     (if (<= d -2.8e-105)
       t_0
       (if (<= d 3.8e-138)
         (- (/ b c) (* d (/ a (pow c 2.0))))
         (if (<= d 1.45e+54) t_0 (/ (- (* c (/ b d)) 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 <= -1.45e+89) {
		tmp = (c * ((b * (1.0 / d)) / d)) - (a / d);
	} else if (d <= -2.8e-105) {
		tmp = t_0;
	} else if (d <= 3.8e-138) {
		tmp = (b / c) - (d * (a / pow(c, 2.0)));
	} else if (d <= 1.45e+54) {
		tmp = t_0;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
    if (d <= (-1.45d+89)) then
        tmp = (c * ((b * (1.0d0 / d)) / d)) - (a / d)
    else if (d <= (-2.8d-105)) then
        tmp = t_0
    else if (d <= 3.8d-138) then
        tmp = (b / c) - (d * (a / (c ** 2.0d0)))
    else if (d <= 1.45d+54) then
        tmp = t_0
    else
        tmp = ((c * (b / d)) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -1.45e+89) {
		tmp = (c * ((b * (1.0 / d)) / d)) - (a / d);
	} else if (d <= -2.8e-105) {
		tmp = t_0;
	} else if (d <= 3.8e-138) {
		tmp = (b / c) - (d * (a / Math.pow(c, 2.0)));
	} else if (d <= 1.45e+54) {
		tmp = t_0;
	} else {
		tmp = ((c * (b / d)) - 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 <= -1.45e+89:
		tmp = (c * ((b * (1.0 / d)) / d)) - (a / d)
	elif d <= -2.8e-105:
		tmp = t_0
	elif d <= 3.8e-138:
		tmp = (b / c) - (d * (a / math.pow(c, 2.0)))
	elif d <= 1.45e+54:
		tmp = t_0
	else:
		tmp = ((c * (b / d)) - 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 <= -1.45e+89)
		tmp = Float64(Float64(c * Float64(Float64(b * Float64(1.0 / d)) / d)) - Float64(a / d));
	elseif (d <= -2.8e-105)
		tmp = t_0;
	elseif (d <= 3.8e-138)
		tmp = Float64(Float64(b / c) - Float64(d * Float64(a / (c ^ 2.0))));
	elseif (d <= 1.45e+54)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - 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 <= -1.45e+89)
		tmp = (c * ((b * (1.0 / d)) / d)) - (a / d);
	elseif (d <= -2.8e-105)
		tmp = t_0;
	elseif (d <= 3.8e-138)
		tmp = (b / c) - (d * (a / (c ^ 2.0)));
	elseif (d <= 1.45e+54)
		tmp = t_0;
	else
		tmp = ((c * (b / d)) - 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, -1.45e+89], N[(N[(c * N[(N[(b * N[(1.0 / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.8e-105], t$95$0, If[LessEqual[d, 3.8e-138], N[(N[(b / c), $MachinePrecision] - N[(d * N[(a / N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.45e+54], t$95$0, N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.45000000000000013e89

   1. Initial program 30.8%

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

   2. Taylor expanded in c around 0 82.4%

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

      1. +-commutative 82.4%

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

      2. mul-1-neg 82.4%

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

      3. unsub-neg 82.4%

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

      4. associate-/l* 82.5%

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

      5. associate-/r/ 80.5%

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

   4. Simplified 80.5%

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

      1. *-un-lft-identity 80.5%

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

      2. pow2 80.5%

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

      3. times-frac 82.6%

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

   6. Applied egg-rr 82.6%

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

      1. associate-*r/ 82.6%

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

   8. Simplified 82.6%

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

   if -1.45000000000000013e89 < d < -2.8e-105 or 3.8000000000000002e-138 < d < 1.4499999999999999e54

   1. Initial program 88.2%

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

   if -2.8e-105 < d < 3.8000000000000002e-138

   1. Initial program 67.4%

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

   2. Taylor expanded in c around inf 85.9%

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

      1. +-commutative 85.9%

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

      2. mul-1-neg 85.9%

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

      3. unsub-neg 85.9%

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

      4. associate-/l* 84.8%

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

      5. associate-/r/ 87.4%

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

   4. Simplified 87.4%

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

   if 1.4499999999999999e54 < d

   1. Initial program 35.0%

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

   2. Taylor expanded in c around 0 79.0%

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

      1. +-commutative 79.0%

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

      2. mul-1-neg 79.0%

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

      3. unsub-neg 79.0%

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

      4. associate-/l* 78.0%

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

      5. associate-/r/ 80.7%

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

   4. Simplified 80.7%

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

      1. *-un-lft-identity 80.7%

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

      2. pow2 80.7%

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

      3. times-frac 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. associate-*l/ 83.7%

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

      2. *-lft-identity 83.7%

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

   8. Simplified 83.7%

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

      1. associate-*l/ 87.9%

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

      2. sub-div 88.0%

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

   10. Applied egg-rr 88.0%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{+89}:\\ \;\;\;\;c \cdot \frac{b \cdot \frac{1}{d}}{d} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-138}:\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{a}{{c}^{2}}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{+54}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]

Alternative 9: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -3.6 \cdot 10^{+88}:\\ \;\;\;\;c \cdot \frac{b \cdot \frac{1}{d}}{d} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-170}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 6.1 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (<= d -3.6e+88)
     (- (* c (/ (* b (/ 1.0 d)) d)) (/ a d))
     (if (<= d -1e-92)
       t_0
       (if (<= d 2.7e-170)
         (/ b c)
         (if (<= d 6.1e+53) t_0 (/ (- (* c (/ b d)) 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 <= -3.6e+88) {
		tmp = (c * ((b * (1.0 / d)) / d)) - (a / d);
	} else if (d <= -1e-92) {
		tmp = t_0;
	} else if (d <= 2.7e-170) {
		tmp = b / c;
	} else if (d <= 6.1e+53) {
		tmp = t_0;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
    if (d <= (-3.6d+88)) then
        tmp = (c * ((b * (1.0d0 / d)) / d)) - (a / d)
    else if (d <= (-1d-92)) then
        tmp = t_0
    else if (d <= 2.7d-170) then
        tmp = b / c
    else if (d <= 6.1d+53) then
        tmp = t_0
    else
        tmp = ((c * (b / d)) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -3.6e+88) {
		tmp = (c * ((b * (1.0 / d)) / d)) - (a / d);
	} else if (d <= -1e-92) {
		tmp = t_0;
	} else if (d <= 2.7e-170) {
		tmp = b / c;
	} else if (d <= 6.1e+53) {
		tmp = t_0;
	} else {
		tmp = ((c * (b / d)) - 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 <= -3.6e+88:
		tmp = (c * ((b * (1.0 / d)) / d)) - (a / d)
	elif d <= -1e-92:
		tmp = t_0
	elif d <= 2.7e-170:
		tmp = b / c
	elif d <= 6.1e+53:
		tmp = t_0
	else:
		tmp = ((c * (b / d)) - 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 <= -3.6e+88)
		tmp = Float64(Float64(c * Float64(Float64(b * Float64(1.0 / d)) / d)) - Float64(a / d));
	elseif (d <= -1e-92)
		tmp = t_0;
	elseif (d <= 2.7e-170)
		tmp = Float64(b / c);
	elseif (d <= 6.1e+53)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - 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 <= -3.6e+88)
		tmp = (c * ((b * (1.0 / d)) / d)) - (a / d);
	elseif (d <= -1e-92)
		tmp = t_0;
	elseif (d <= 2.7e-170)
		tmp = b / c;
	elseif (d <= 6.1e+53)
		tmp = t_0;
	else
		tmp = ((c * (b / d)) - 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, -3.6e+88], N[(N[(c * N[(N[(b * N[(1.0 / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-92], t$95$0, If[LessEqual[d, 2.7e-170], N[(b / c), $MachinePrecision], If[LessEqual[d, 6.1e+53], t$95$0, N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.6000000000000002e88

   1. Initial program 30.8%

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

   2. Taylor expanded in c around 0 82.4%

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

      1. +-commutative 82.4%

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

      2. mul-1-neg 82.4%

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

      3. unsub-neg 82.4%

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

      4. associate-/l* 82.5%

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

      5. associate-/r/ 80.5%

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

   4. Simplified 80.5%

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

      1. *-un-lft-identity 80.5%

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

      2. pow2 80.5%

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

      3. times-frac 82.6%

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

   6. Applied egg-rr 82.6%

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

      1. associate-*r/ 82.6%

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

   8. Simplified 82.6%

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

   if -3.6000000000000002e88 < d < -9.99999999999999988e-93 or 2.6999999999999999e-170 < d < 6.1000000000000002e53

   1. Initial program 87.8%

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

   if -9.99999999999999988e-93 < d < 2.6999999999999999e-170

   1. Initial program 65.9%

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

   2. Taylor expanded in c around inf 75.6%

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

   if 6.1000000000000002e53 < d

   1. Initial program 35.0%

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

   2. Taylor expanded in c around 0 79.0%

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

      1. +-commutative 79.0%

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

      2. mul-1-neg 79.0%

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

      3. unsub-neg 79.0%

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

      4. associate-/l* 78.0%

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

      5. associate-/r/ 80.7%

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

   4. Simplified 80.7%

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

      1. *-un-lft-identity 80.7%

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

      2. pow2 80.7%

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

      3. times-frac 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. associate-*l/ 83.7%

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

      2. *-lft-identity 83.7%

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

   8. Simplified 83.7%

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

      1. associate-*l/ 87.9%

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

      2. sub-div 88.0%

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

   10. Applied egg-rr 88.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.6 \cdot 10^{+88}:\\ \;\;\;\;c \cdot \frac{b \cdot \frac{1}{d}}{d} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-92}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-170}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 6.1 \cdot 10^{+53}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]

Alternative 10: 71.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.45 \cdot 10^{+23} \lor \neg \left(c \leq 4.7 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{\frac{c}{d}}{d} - \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.45e+23) (not (<= c 4.7e+14)))
   (/ b c)
   (- (* b (/ (/ c d) d)) (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.45e+23) || !(c <= 4.7e+14)) {
		tmp = b / c;
	} else {
		tmp = (b * ((c / d) / d)) - (a / d);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-1.45d+23)) .or. (.not. (c <= 4.7d+14))) then
        tmp = b / c
    else
        tmp = (b * ((c / d) / d)) - (a / d)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.45e+23) || !(c <= 4.7e+14)) {
		tmp = b / c;
	} else {
		tmp = (b * ((c / d) / d)) - (a / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.45e+23) or not (c <= 4.7e+14):
		tmp = b / c
	else:
		tmp = (b * ((c / d) / d)) - (a / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.45e+23) || !(c <= 4.7e+14))
		tmp = Float64(b / c);
	else
		tmp = Float64(Float64(b * Float64(Float64(c / d) / d)) - Float64(a / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.45e+23) || ~((c <= 4.7e+14)))
		tmp = b / c;
	else
		tmp = (b * ((c / d) / d)) - (a / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.45e+23], N[Not[LessEqual[c, 4.7e+14]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(N[(b * N[(N[(c / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.45000000000000006e23 or 4.7e14 < c

   1. Initial program 51.6%

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

   2. Taylor expanded in c around inf 62.8%

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

   if -1.45000000000000006e23 < c < 4.7e14

   1. Initial program 64.5%

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

      1. div-sub 60.2%

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

      2. sub-neg 60.2%

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

      3. *-commutative 60.2%

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

      4. add-sqr-sqrt 60.2%

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

      5. times-frac 58.5%

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

      6. fma-def 58.5%

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

      7. hypot-def 58.5%

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

      8. hypot-def 59.9%

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

      9. associate-/l* 66.7%

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

      10. add-sqr-sqrt 66.7%

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

      11. pow2 66.7%

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

      12. hypot-def 66.7%

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

   3. Applied egg-rr 66.7%

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

   4. Taylor expanded in c around 0 82.6%

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

      1. neg-mul-1 82.6%

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

      2. +-commutative 82.6%

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

      3. unsub-neg 82.6%

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

      4. *-commutative 82.6%

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

      5. associate-/l* 80.4%

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

      6. associate-/r/ 82.9%

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

   6. Simplified 82.9%

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

      1. *-un-lft-identity 82.9%

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

      2. unpow2 82.9%

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

      3. times-frac 84.2%

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

   8. Applied egg-rr 84.2%

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

      1. associate-*l/ 84.2%

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

      2. *-lft-identity 84.2%

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

   10. Simplified 84.2%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.45 \cdot 10^{+23} \lor \neg \left(c \leq 4.7 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{\frac{c}{d}}{d} - \frac{a}{d}\\ \end{array} \]

Alternative 11: 72.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -8.5 \cdot 10^{+22} \lor \neg \left(c \leq 2.8 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -8.5e+22) (not (<= c 2.8e+14)))
   (/ b c)
   (/ (- (* c (/ b d)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -8.5e+22) || !(c <= 2.8e+14)) {
		tmp = b / c;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-8.5d+22)) .or. (.not. (c <= 2.8d+14))) then
        tmp = b / c
    else
        tmp = ((c * (b / d)) - 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 <= -8.5e+22) || !(c <= 2.8e+14)) {
		tmp = b / c;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -8.5e+22) or not (c <= 2.8e+14):
		tmp = b / c
	else:
		tmp = ((c * (b / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -8.5e+22) || !(c <= 2.8e+14))
		tmp = Float64(b / c);
	else
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -8.5e+22) || ~((c <= 2.8e+14)))
		tmp = b / c;
	else
		tmp = ((c * (b / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -8.5e+22], N[Not[LessEqual[c, 2.8e+14]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -8.49999999999999979e22 or 2.8e14 < c

   1. Initial program 51.6%

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

   2. Taylor expanded in c around inf 62.8%

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

   if -8.49999999999999979e22 < c < 2.8e14

   1. Initial program 64.5%

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

   2. Taylor expanded in c around 0 82.6%

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

      1. +-commutative 82.6%

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

      2. mul-1-neg 82.6%

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

      3. unsub-neg 82.6%

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

      4. associate-/l* 82.8%

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

      5. associate-/r/ 79.5%

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

   4. Simplified 79.5%

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

      1. *-un-lft-identity 79.5%

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

      2. pow2 79.5%

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

      3. times-frac 80.1%

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

   6. Applied egg-rr 80.1%

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

      1. associate-*l/ 80.1%

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

      2. *-lft-identity 80.1%

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

   8. Simplified 80.1%

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

      1. associate-*l/ 83.6%

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

      2. sub-div 83.6%

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

   10. Applied egg-rr 83.6%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8.5 \cdot 10^{+22} \lor \neg \left(c \leq 2.8 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]

Alternative 12: 63.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{-63} \lor \neg \left(d \leq 1.05 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.5e-63) (not (<= d 1.05e-11))) (/ (- a) d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.5e-63) || !(d <= 1.05e-11)) {
		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 <= (-1.5d-63)) .or. (.not. (d <= 1.05d-11))) 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 <= -1.5e-63) || !(d <= 1.05e-11)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.5e-63) or not (d <= 1.05e-11):
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.5e-63) || !(d <= 1.05e-11))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.5e-63) || ~((d <= 1.05e-11)))
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.5e-63], N[Not[LessEqual[d, 1.05e-11]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.4999999999999999e-63 or 1.0499999999999999e-11 < d

   1. Initial program 48.8%

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

   2. Taylor expanded in c around 0 68.3%

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

      1. associate-*r/ 68.3%

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

      2. neg-mul-1 68.3%

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

   4. Simplified 68.3%

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

   if -1.4999999999999999e-63 < d < 1.0499999999999999e-11

   1. Initial program 74.9%

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

   2. Taylor expanded in c around inf 67.6%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{-63} \lor \neg \left(d \leq 1.05 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 13: 9.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.8%

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

2. Taylor expanded in b around 0 38.1%

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

   1. associate-*r/ 38.1%

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

   2. associate-*r* 38.1%

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

   3. neg-mul-1 38.1%

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

   4. *-commutative 38.1%

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

   5. associate-/l* 40.2%

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

   6. +-commutative 40.2%

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

   7. unpow2 40.2%

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

   8. fma-udef 40.2%

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

4. Simplified 40.2%

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

   1. fma-udef 40.2%

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

   2. +-commutative 40.2%

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

   3. add-sqr-sqrt 40.2%

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

   4. pow2 40.2%

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

   5. hypot-udef 40.2%

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

   6. pow2 40.2%

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

   7. hypot-udef 40.2%

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

   8. neg-mul-1 40.2%

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

   9. times-frac 49.2%

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

6. Applied egg-rr 49.2%

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

7. Taylor expanded in d around -inf 29.0%

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

   1. associate-*r/ 29.0%

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

   2. neg-mul-1 29.0%

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

9. Simplified 29.0%

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

10. Taylor expanded in d around 0 10.5%

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

11. Final simplification 10.5%

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

Alternative 14: 42.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.8%

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

2. Taylor expanded in c around inf 36.1%

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

3. Final simplification 36.1%

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

Developer target: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023315 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d)))))

  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))