Complex division, imag part

Percentage Accurate: 61.7% → 96.2%

Time: 12.7s

Alternatives: 13

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.7% 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: 96.2% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

   1. fma-neg 62.6%

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

   2. distribute-rgt-neg-out 62.6%

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

   3. +-commutative 62.6%

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

   4. fma-define 62.6%

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

3. Simplified 62.6%

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

   1. distribute-rgt-neg-out 62.6%

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

   2. fma-neg 62.6%

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

   3. fma-undefine 62.6%

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

   4. +-commutative 62.6%

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

   5. div-sub 58.6%

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

   6. *-commutative 58.6%

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

   7. add-sqr-sqrt 58.6%

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

   8. times-frac 60.3%

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

   9. fma-neg 60.3%

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

   10. hypot-define 60.3%

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

   11. hypot-define 73.7%

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

   12. associate-/l* 76.7%

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

   13. add-sqr-sqrt 76.7%

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

   14. pow2 76.7%

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

6. Applied egg-rr 76.7%

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

   1. *-un-lft-identity 76.7%

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

   2. unpow2 76.7%

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

   3. times-frac 95.5%

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

8. Applied egg-rr 95.5%

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

   1. associate-*l/ 95.5%

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

   2. *-lft-identity 95.5%

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

   3. hypot-undefine 76.7%

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

   4. unpow2 76.7%

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

   5. unpow2 76.7%

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

   6. +-commutative 76.7%

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

   7. unpow2 76.7%

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

   8. unpow2 76.7%

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

   9. hypot-define 95.5%

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

   10. hypot-undefine 76.7%

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

   11. unpow2 76.7%

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

   12. unpow2 76.7%

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

   13. +-commutative 76.7%

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

   14. unpow2 76.7%

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

   15. unpow2 76.7%

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

   16. hypot-define 95.5%

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

10. Simplified 95.5%

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

11. Final simplification 95.5%

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

Alternative 2: 93.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ t_2 := \mathsf{fma}\left(t\_0, t\_1, a \cdot \frac{d}{-{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ t_3 := \mathsf{fma}\left(t\_0, t\_1, \frac{a}{-d}\right)\\ \mathbf{if}\;d \leq -5.5 \cdot 10^{+111}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-120}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-168}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 6.9 \cdot 10^{+136}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 (/ d (- (pow (hypot c d) 2.0))))))
        (t_3 (fma t_0 t_1 (/ a (- d)))))
   (if (<= d -5.5e+111)
     t_3
     (if (<= d -1.6e-120)
       t_2
       (if (<= d 1.6e-168)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 6.9e+136) t_2 t_3))))))

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 * (d / -pow(hypot(c, d), 2.0))));
	double t_3 = fma(t_0, t_1, (a / -d));
	double tmp;
	if (d <= -5.5e+111) {
		tmp = t_3;
	} else if (d <= -1.6e-120) {
		tmp = t_2;
	} else if (d <= 1.6e-168) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 6.9e+136) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	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(a * Float64(d / Float64(-(hypot(c, d) ^ 2.0)))))
	t_3 = fma(t_0, t_1, Float64(a / Float64(-d)))
	tmp = 0.0
	if (d <= -5.5e+111)
		tmp = t_3;
	elseif (d <= -1.6e-120)
		tmp = t_2;
	elseif (d <= 1.6e-168)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 6.9e+136)
		tmp = t_2;
	else
		tmp = t_3;
	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[(d / (-N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * t$95$1 + N[(a / (-d)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.5e+111], t$95$3, If[LessEqual[d, -1.6e-120], t$95$2, If[LessEqual[d, 1.6e-168], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 6.9e+136], t$95$2, t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.4999999999999998e111 or 6.9e136 < d

   1. Initial program 36.2%

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

      1. fma-neg 36.2%

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

      2. distribute-rgt-neg-out 36.2%

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

      3. +-commutative 36.2%

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

      4. fma-define 36.2%

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

   3. Simplified 36.2%

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

      1. distribute-rgt-neg-out 36.2%

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

      2. fma-neg 36.2%

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

      3. fma-undefine 36.2%

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

      4. +-commutative 36.2%

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

      5. div-sub 36.2%

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

      6. *-commutative 36.2%

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

      7. add-sqr-sqrt 36.2%

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

      8. times-frac 36.5%

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

      9. fma-neg 36.5%

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

      10. hypot-define 36.5%

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

      11. hypot-define 45.1%

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

      12. associate-/l* 50.0%

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

      13. add-sqr-sqrt 50.0%

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

      14. pow2 50.0%

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

   6. Applied egg-rr 50.0%

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

   7. Taylor expanded in d around inf 96.4%

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

   if -5.4999999999999998e111 < d < -1.6e-120 or 1.60000000000000003e-168 < d < 6.9e136

   1. Initial program 77.8%

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

      1. fma-neg 77.8%

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

      2. distribute-rgt-neg-out 77.8%

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

      3. +-commutative 77.8%

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

      4. fma-define 77.8%

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

   3. Simplified 77.8%

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

      1. distribute-rgt-neg-out 77.8%

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

      2. fma-neg 77.8%

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

      3. fma-undefine 77.8%

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

      4. +-commutative 77.8%

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

      5. div-sub 76.9%

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

      6. *-commutative 76.9%

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

      7. add-sqr-sqrt 76.9%

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

      8. times-frac 77.5%

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

      9. fma-neg 77.5%

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

      10. hypot-define 77.5%

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

      11. hypot-define 90.1%

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

      12. associate-/l* 94.7%

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

      13. add-sqr-sqrt 94.7%

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

      14. pow2 94.7%

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

   6. Applied egg-rr 94.7%

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

   if -1.6e-120 < d < 1.60000000000000003e-168

   1. Initial program 68.4%

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

      1. fma-neg 68.4%

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

      2. distribute-rgt-neg-out 68.4%

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

      3. +-commutative 68.4%

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

      4. fma-define 68.4%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in c around inf 91.8%

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

      1. associate-*r/ 91.8%

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

      2. neg-mul-1 91.8%

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

      3. distribute-lft-neg-in 91.8%

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

   7. Simplified 91.8%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{a}{-d}\right)\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-120}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{d}{-{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-168}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 6.9 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{d}{-{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{a}{-d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{a}{-d}\right)\\ \mathbf{if}\;d \leq -1.55 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{-117}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 10^{+29}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma (/ c (hypot c d)) (/ b (hypot c d)) (/ a (- d)))))
   (if (<= d -1.55e-34)
     t_0
     (if (<= d 4.8e-117)
       (/ (- b (/ (* d a) c)) c)
       (if (<= d 1e+29) (/ (- (* c b) (* d a)) (+ (* c c) (* d d))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((c / hypot(c, d)), (b / hypot(c, d)), (a / -d));
	double tmp;
	if (d <= -1.55e-34) {
		tmp = t_0;
	} else if (d <= 4.8e-117) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 1e+29) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(a / Float64(-d)))
	tmp = 0.0
	if (d <= -1.55e-34)
		tmp = t_0;
	elseif (d <= 4.8e-117)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 1e+29)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[(a / (-d)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.55e-34], t$95$0, If[LessEqual[d, 4.8e-117], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1e+29], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.5499999999999999e-34 or 9.99999999999999914e28 < d

   1. Initial program 50.8%

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

      1. fma-neg 50.8%

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

      2. distribute-rgt-neg-out 50.8%

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

      3. +-commutative 50.8%

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

      4. fma-define 50.8%

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

   3. Simplified 50.8%

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

      1. distribute-rgt-neg-out 50.8%

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

      2. fma-neg 50.8%

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

      3. fma-undefine 50.8%

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

      4. +-commutative 50.8%

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

      5. div-sub 50.8%

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

      6. *-commutative 50.8%

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

      7. add-sqr-sqrt 50.9%

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

      8. times-frac 52.7%

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

      9. fma-neg 52.7%

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

      10. hypot-define 52.7%

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

      11. hypot-define 61.9%

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

      12. associate-/l* 67.3%

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

      13. add-sqr-sqrt 67.3%

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

      14. pow2 67.3%

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

   6. Applied egg-rr 67.3%

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

   7. Taylor expanded in d around inf 91.4%

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

   if -1.5499999999999999e-34 < d < 4.80000000000000028e-117

   1. Initial program 68.9%

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

      1. fma-neg 68.9%

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

      2. distribute-rgt-neg-out 68.9%

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

      3. +-commutative 68.9%

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

      4. fma-define 68.9%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in c around inf 91.0%

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

      1. associate-*r/ 91.0%

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

      2. neg-mul-1 91.0%

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

      3. distribute-lft-neg-in 91.0%

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

   7. Simplified 91.0%

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

   if 4.80000000000000028e-117 < d < 9.99999999999999914e28

   1. Initial program 87.6%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{a}{-d}\right)\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{-117}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 10^{+29}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{a}{-d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.1 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{-d}, \frac{a}{-d}\right)\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-113}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{\frac{c}{d}}{d} - \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -3.1e-12)
   (fma (/ c (hypot c d)) (/ b (- d)) (/ a (- d)))
   (if (<= d 1.7e-113)
     (/ (- b (/ (* d a) c)) c)
     (if (<= d 1.8e+29)
       (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))
       (- (* b (/ (/ c d) d)) (/ a d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.1e-12) {
		tmp = fma((c / hypot(c, d)), (b / -d), (a / -d));
	} else if (d <= 1.7e-113) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 1.8e+29) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else {
		tmp = (b * ((c / d) / d)) - (a / d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -3.1e-12)
		tmp = fma(Float64(c / hypot(c, d)), Float64(b / Float64(-d)), Float64(a / Float64(-d)));
	elseif (d <= 1.7e-113)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 1.8e+29)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(Float64(b * Float64(Float64(c / d) / d)) - Float64(a / d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -3.1e-12], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / (-d)), $MachinePrecision] + N[(a / (-d)), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.7e-113], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.8e+29], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(N[(c / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.1000000000000001e-12

   1. Initial program 46.1%

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

      1. fma-neg 46.1%

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

      2. distribute-rgt-neg-out 46.1%

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

      3. +-commutative 46.1%

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

      4. fma-define 46.2%

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

   3. Simplified 46.2%

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

      1. distribute-rgt-neg-out 46.2%

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

      2. fma-neg 46.2%

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

      3. fma-undefine 46.1%

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

      4. +-commutative 46.1%

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

      5. div-sub 46.1%

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

      6. *-commutative 46.1%

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

      7. add-sqr-sqrt 46.2%

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

      8. times-frac 49.4%

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

      9. fma-neg 49.4%

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

      10. hypot-define 49.4%

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

      11. hypot-define 62.1%

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

      12. associate-/l* 64.3%

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

      13. add-sqr-sqrt 64.3%

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

      14. pow2 64.3%

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

   6. Applied egg-rr 64.3%

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

   7. Taylor expanded in d around inf 91.3%

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

   8. Taylor expanded in d around -inf 84.2%

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

      1. associate-*r/ 84.2%

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

      2. neg-mul-1 84.2%

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

   10. Simplified 84.2%

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

   if -3.1000000000000001e-12 < d < 1.7000000000000001e-113

   1. Initial program 69.4%

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

      1. fma-neg 69.4%

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

      2. distribute-rgt-neg-out 69.4%

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

      3. +-commutative 69.4%

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

      4. fma-define 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in c around inf 89.4%

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

      1. associate-*r/ 89.4%

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

      2. neg-mul-1 89.4%

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

      3. distribute-lft-neg-in 89.4%

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

   7. Simplified 89.4%

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

   if 1.7000000000000001e-113 < d < 1.79999999999999988e29

   1. Initial program 87.6%

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

   if 1.79999999999999988e29 < d

   1. Initial program 54.2%

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

      1. fma-neg 54.2%

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

      2. distribute-rgt-neg-out 54.2%

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

      3. +-commutative 54.2%

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

      4. fma-define 54.1%

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

   3. Simplified 54.1%

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

      1. distribute-rgt-neg-out 54.1%

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

      2. fma-neg 54.1%

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

      3. fma-undefine 54.2%

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

      4. +-commutative 54.2%

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

      5. div-sub 54.2%

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

      6. *-commutative 54.2%

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

      7. add-sqr-sqrt 54.2%

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

      8. times-frac 54.4%

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

      9. fma-neg 54.4%

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

      10. hypot-define 54.4%

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

      11. hypot-define 58.0%

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

      12. associate-/l* 68.0%

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

      13. add-sqr-sqrt 68.0%

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

      14. pow2 68.0%

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

   6. Applied egg-rr 68.0%

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

   7. Taylor expanded in c around 0 89.6%

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

      1. associate-/l* 89.7%

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

   9. Simplified 89.7%

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

       1. *-un-lft-identity 89.7%

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

       2. unpow2 89.7%

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

       3. times-frac 89.9%

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

   11. Applied egg-rr 89.9%

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

       1. associate-*l/ 89.9%

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

       2. *-lft-identity 89.9%

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

   13. Simplified 89.9%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.1 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{-d}, \frac{a}{-d}\right)\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-113}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{\frac{c}{d}}{d} - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{\frac{c}{d}}{d} - \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -5e-10)
   (/ (- (* c (/ b d)) a) d)
   (if (<= d 1.2e-117)
     (/ (- b (/ (* d a) c)) c)
     (if (<= d 1.8e+29)
       (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))
       (- (* b (/ (/ c d) d)) (/ a d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -5e-10) {
		tmp = ((c * (b / d)) - a) / d;
	} else if (d <= 1.2e-117) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 1.8e+29) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} 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 (d <= (-5d-10)) then
        tmp = ((c * (b / d)) - a) / d
    else if (d <= 1.2d-117) then
        tmp = (b - ((d * a) / c)) / c
    else if (d <= 1.8d+29) then
        tmp = ((c * b) - (d * a)) / ((c * c) + (d * d))
    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 (d <= -5e-10) {
		tmp = ((c * (b / d)) - a) / d;
	} else if (d <= 1.2e-117) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 1.8e+29) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else {
		tmp = (b * ((c / d) / d)) - (a / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -5e-10:
		tmp = ((c * (b / d)) - a) / d
	elif d <= 1.2e-117:
		tmp = (b - ((d * a) / c)) / c
	elif d <= 1.8e+29:
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d))
	else:
		tmp = (b * ((c / d) / d)) - (a / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -5e-10)
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	elseif (d <= 1.2e-117)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 1.8e+29)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	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 (d <= -5e-10)
		tmp = ((c * (b / d)) - a) / d;
	elseif (d <= 1.2e-117)
		tmp = (b - ((d * a) / c)) / c;
	elseif (d <= 1.8e+29)
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	else
		tmp = (b * ((c / d) / d)) - (a / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -5e-10], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 1.2e-117], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.8e+29], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(N[(c / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.00000000000000031e-10

   1. Initial program 46.1%

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

      1. fma-neg 46.1%

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

      2. distribute-rgt-neg-out 46.1%

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

      3. +-commutative 46.1%

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

      4. fma-define 46.2%

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

   3. Simplified 46.2%

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

   5. Taylor expanded in c around 0 75.1%

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

      1. +-commutative 75.1%

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

      2. mul-1-neg 75.1%

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

      3. unsub-neg 75.1%

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

      4. unpow2 75.1%

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

      5. associate-/r* 76.7%

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

      6. div-sub 76.7%

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

      7. *-commutative 76.7%

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

      8. associate-/l* 83.9%

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

   7. Simplified 83.9%

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

   if -5.00000000000000031e-10 < d < 1.20000000000000007e-117

   1. Initial program 69.4%

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

      1. fma-neg 69.4%

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

      2. distribute-rgt-neg-out 69.4%

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

      3. +-commutative 69.4%

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

      4. fma-define 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in c around inf 89.4%

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

      1. associate-*r/ 89.4%

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

      2. neg-mul-1 89.4%

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

      3. distribute-lft-neg-in 89.4%

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

   7. Simplified 89.4%

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

   if 1.20000000000000007e-117 < d < 1.79999999999999988e29

   1. Initial program 87.6%

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

   if 1.79999999999999988e29 < d

   1. Initial program 54.2%

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

      1. fma-neg 54.2%

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

      2. distribute-rgt-neg-out 54.2%

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

      3. +-commutative 54.2%

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

      4. fma-define 54.1%

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

   3. Simplified 54.1%

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

      1. distribute-rgt-neg-out 54.1%

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

      2. fma-neg 54.1%

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

      3. fma-undefine 54.2%

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

      4. +-commutative 54.2%

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

      5. div-sub 54.2%

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

      6. *-commutative 54.2%

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

      7. add-sqr-sqrt 54.2%

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

      8. times-frac 54.4%

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

      9. fma-neg 54.4%

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

      10. hypot-define 54.4%

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

      11. hypot-define 58.0%

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

      12. associate-/l* 68.0%

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

      13. add-sqr-sqrt 68.0%

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

      14. pow2 68.0%

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

   6. Applied egg-rr 68.0%

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

   7. Taylor expanded in c around 0 89.6%

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

      1. associate-/l* 89.7%

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

   9. Simplified 89.7%

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

       1. *-un-lft-identity 89.7%

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

       2. unpow2 89.7%

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

       3. times-frac 89.9%

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

   11. Applied egg-rr 89.9%

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

       1. associate-*l/ 89.9%

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

       2. *-lft-identity 89.9%

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

   13. Simplified 89.9%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{\frac{c}{d}}{d} - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{-7}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{\frac{c}{d}}{d} - \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -5.5e-9)
   (/ (- (* c (/ b d)) a) d)
   (if (<= d 1.02e-7)
     (/ (- b (/ (* d a) c)) c)
     (- (* b (/ (/ c d) d)) (/ a d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -5.5e-9) {
		tmp = ((c * (b / d)) - a) / d;
	} else if (d <= 1.02e-7) {
		tmp = (b - ((d * a) / c)) / 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 (d <= (-5.5d-9)) then
        tmp = ((c * (b / d)) - a) / d
    else if (d <= 1.02d-7) then
        tmp = (b - ((d * a) / c)) / 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 (d <= -5.5e-9) {
		tmp = ((c * (b / d)) - a) / d;
	} else if (d <= 1.02e-7) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = (b * ((c / d) / d)) - (a / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -5.5e-9:
		tmp = ((c * (b / d)) - a) / d
	elif d <= 1.02e-7:
		tmp = (b - ((d * a) / c)) / c
	else:
		tmp = (b * ((c / d) / d)) - (a / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -5.5e-9)
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	elseif (d <= 1.02e-7)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / 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 (d <= -5.5e-9)
		tmp = ((c * (b / d)) - a) / d;
	elseif (d <= 1.02e-7)
		tmp = (b - ((d * a) / c)) / c;
	else
		tmp = (b * ((c / d) / d)) - (a / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -5.5e-9], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 1.02e-7], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b * N[(N[(c / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.4999999999999996e-9

   1. Initial program 46.1%

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

      1. fma-neg 46.1%

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

      2. distribute-rgt-neg-out 46.1%

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

      3. +-commutative 46.1%

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

      4. fma-define 46.2%

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

   3. Simplified 46.2%

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

   5. Taylor expanded in c around 0 75.1%

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

      1. +-commutative 75.1%

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

      2. mul-1-neg 75.1%

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

      3. unsub-neg 75.1%

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

      4. unpow2 75.1%

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

      5. associate-/r* 76.7%

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

      6. div-sub 76.7%

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

      7. *-commutative 76.7%

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

      8. associate-/l* 83.9%

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

   7. Simplified 83.9%

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

   if -5.4999999999999996e-9 < d < 1.02e-7

   1. Initial program 72.0%

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

      1. fma-neg 72.0%

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

      2. distribute-rgt-neg-out 72.0%

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

      3. +-commutative 72.0%

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

      4. fma-define 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in c around inf 84.6%

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

      1. associate-*r/ 84.6%

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

      2. neg-mul-1 84.6%

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

      3. distribute-lft-neg-in 84.6%

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

   7. Simplified 84.6%

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

   if 1.02e-7 < d

   1. Initial program 60.8%

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

      1. fma-neg 60.8%

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

      2. distribute-rgt-neg-out 60.8%

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

      3. +-commutative 60.8%

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

      4. fma-define 60.8%

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

   3. Simplified 60.8%

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

      1. distribute-rgt-neg-out 60.8%

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

      2. fma-neg 60.8%

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

      3. fma-undefine 60.8%

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

      4. +-commutative 60.8%

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

      5. div-sub 60.8%

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

      6. *-commutative 60.8%

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

      7. add-sqr-sqrt 60.8%

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

      8. times-frac 61.0%

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

      9. fma-neg 61.0%

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

      10. hypot-define 61.0%

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

      11. hypot-define 64.1%

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

      12. associate-/l* 72.6%

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

      13. add-sqr-sqrt 72.6%

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

      14. pow2 72.6%

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

   6. Applied egg-rr 72.6%

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

   7. Taylor expanded in c around 0 87.9%

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

      1. associate-/l* 88.1%

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

   9. Simplified 88.1%

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

       1. *-un-lft-identity 88.1%

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

       2. unpow2 88.1%

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

       3. times-frac 88.2%

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

   11. Applied egg-rr 88.2%

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

       1. associate-*l/ 88.2%

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

       2. *-lft-identity 88.2%

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

   13. Simplified 88.2%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{-7}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{\frac{c}{d}}{d} - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{-11} \lor \neg \left(d \leq 5.2 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -5.8e-11) (not (<= d 5.2e-8)))
   (/ (- (* c (/ b d)) a) d)
   (/ (- b (/ (* d a) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.8e-11) || !(d <= 5.2e-8)) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = (b - ((d * a) / c)) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-5.8d-11)) .or. (.not. (d <= 5.2d-8))) then
        tmp = ((c * (b / d)) - a) / d
    else
        tmp = (b - ((d * a) / c)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.8e-11) || !(d <= 5.2e-8)) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = (b - ((d * a) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -5.8e-11) or not (d <= 5.2e-8):
		tmp = ((c * (b / d)) - a) / d
	else:
		tmp = (b - ((d * a) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -5.8e-11) || !(d <= 5.2e-8))
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -5.8e-11) || ~((d <= 5.2e-8)))
		tmp = ((c * (b / d)) - a) / d;
	else
		tmp = (b - ((d * a) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -5.8e-11], N[Not[LessEqual[d, 5.2e-8]], $MachinePrecision]], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -5.8e-11 or 5.2000000000000002e-8 < d

   1. Initial program 53.1%

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

      1. fma-neg 53.1%

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

      2. distribute-rgt-neg-out 53.1%

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

      3. +-commutative 53.1%

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

      4. fma-define 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in c around 0 81.2%

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

      1. +-commutative 81.2%

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

      2. mul-1-neg 81.2%

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

      3. unsub-neg 81.2%

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

      4. unpow2 81.2%

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

      5. associate-/r* 82.1%

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

      6. div-sub 82.1%

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

      7. *-commutative 82.1%

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

      8. associate-/l* 85.9%

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

   7. Simplified 85.9%

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

   if -5.8e-11 < d < 5.2000000000000002e-8

   1. Initial program 72.0%

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

      1. fma-neg 72.0%

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

      2. distribute-rgt-neg-out 72.0%

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

      3. +-commutative 72.0%

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

      4. fma-define 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in c around inf 84.6%

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

      1. associate-*r/ 84.6%

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

      2. neg-mul-1 84.6%

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

      3. distribute-lft-neg-in 84.6%

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

   7. Simplified 84.6%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{-11} \lor \neg \left(d \leq 5.2 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.4 \cdot 10^{-9} \lor \neg \left(d \leq 4.8 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.4e-9) (not (<= d 4.8e-8)))
   (/ (- (* c (/ b d)) a) d)
   (/ (- b (* a (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.4e-9) || !(d <= 4.8e-8)) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-1.4d-9)) .or. (.not. (d <= 4.8d-8))) then
        tmp = ((c * (b / d)) - a) / d
    else
        tmp = (b - (a * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.4e-9) || !(d <= 4.8e-8)) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.4e-9) or not (d <= 4.8e-8):
		tmp = ((c * (b / d)) - a) / d
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.4e-9) || !(d <= 4.8e-8))
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.4e-9) || ~((d <= 4.8e-8)))
		tmp = ((c * (b / d)) - a) / d;
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.4e-9], N[Not[LessEqual[d, 4.8e-8]], $MachinePrecision]], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.39999999999999992e-9 or 4.79999999999999997e-8 < d

   1. Initial program 53.1%

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

      1. fma-neg 53.1%

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

      2. distribute-rgt-neg-out 53.1%

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

      3. +-commutative 53.1%

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

      4. fma-define 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in c around 0 81.2%

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

      1. +-commutative 81.2%

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

      2. mul-1-neg 81.2%

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

      3. unsub-neg 81.2%

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

      4. unpow2 81.2%

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

      5. associate-/r* 82.1%

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

      6. div-sub 82.1%

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

      7. *-commutative 82.1%

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

      8. associate-/l* 85.9%

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

   7. Simplified 85.9%

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

   if -1.39999999999999992e-9 < d < 4.79999999999999997e-8

   1. Initial program 72.0%

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

      1. fma-neg 72.0%

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

      2. distribute-rgt-neg-out 72.0%

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

      3. +-commutative 72.0%

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

      4. fma-define 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in c around inf 84.6%

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

      1. mul-1-neg 84.6%

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

      2. unsub-neg 84.6%

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

      3. associate-/l* 83.5%

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

   7. Simplified 83.5%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.4 \cdot 10^{-9} \lor \neg \left(d \leq 4.8 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{-11} \lor \neg \left(d \leq 5.5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -8.2e-11) (not (<= d 5.5e-8)))
   (/ (- (* b (/ c d)) a) d)
   (/ (- b (* a (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8.2e-11) || !(d <= 5.5e-8)) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-8.2d-11)) .or. (.not. (d <= 5.5d-8))) then
        tmp = ((b * (c / d)) - a) / d
    else
        tmp = (b - (a * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8.2e-11) || !(d <= 5.5e-8)) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -8.2e-11) or not (d <= 5.5e-8):
		tmp = ((b * (c / d)) - a) / d
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -8.2e-11) || !(d <= 5.5e-8))
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -8.2e-11) || ~((d <= 5.5e-8)))
		tmp = ((b * (c / d)) - a) / d;
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -8.2e-11], N[Not[LessEqual[d, 5.5e-8]], $MachinePrecision]], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -8.2000000000000001e-11 or 5.5000000000000003e-8 < d

   1. Initial program 53.1%

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

      1. fma-neg 53.1%

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

      2. distribute-rgt-neg-out 53.1%

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

      3. +-commutative 53.1%

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

      4. fma-define 53.1%

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

   3. Simplified 53.1%

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

      1. distribute-rgt-neg-out 53.1%

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

      2. fma-neg 53.1%

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

      3. fma-undefine 53.1%

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

      4. +-commutative 53.1%

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

      5. div-sub 53.1%

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

      6. *-commutative 53.1%

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

      7. add-sqr-sqrt 53.2%

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

      8. times-frac 54.9%

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

      9. fma-neg 54.9%

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

      10. hypot-define 54.9%

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

      11. hypot-define 63.1%

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

      12. associate-/l* 68.3%

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

      13. add-sqr-sqrt 68.3%

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

      14. pow2 68.3%

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

   6. Applied egg-rr 68.3%

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

   7. Taylor expanded in d around inf 82.1%

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

      1. associate-/l* 85.2%

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

   9. Simplified 85.2%

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

   if -8.2000000000000001e-11 < d < 5.5000000000000003e-8

   1. Initial program 72.0%

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

      1. fma-neg 72.0%

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

      2. distribute-rgt-neg-out 72.0%

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

      3. +-commutative 72.0%

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

      4. fma-define 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in c around inf 84.6%

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

      1. mul-1-neg 84.6%

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

      2. unsub-neg 84.6%

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

      3. associate-/l* 83.5%

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

   7. Simplified 83.5%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{-11} \lor \neg \left(d \leq 5.5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.6 \cdot 10^{-10} \lor \neg \left(d \leq 3.4 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -5.6e-10) (not (<= d 3.4e-7)))
   (/ a (- d))
   (/ (- b (* a (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.6e-10) || !(d <= 3.4e-7)) {
		tmp = a / -d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-5.6d-10)) .or. (.not. (d <= 3.4d-7))) then
        tmp = a / -d
    else
        tmp = (b - (a * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.6e-10) || !(d <= 3.4e-7)) {
		tmp = a / -d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -5.6e-10) or not (d <= 3.4e-7):
		tmp = a / -d
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -5.6e-10) || !(d <= 3.4e-7))
		tmp = Float64(a / Float64(-d));
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -5.6e-10) || ~((d <= 3.4e-7)))
		tmp = a / -d;
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -5.6e-10], N[Not[LessEqual[d, 3.4e-7]], $MachinePrecision]], N[(a / (-d)), $MachinePrecision], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -5.60000000000000031e-10 or 3.39999999999999974e-7 < d

   1. Initial program 53.1%

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

      1. fma-neg 53.1%

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

      2. distribute-rgt-neg-out 53.1%

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

      3. +-commutative 53.1%

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

      4. fma-define 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in c around 0 71.0%

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

      1. associate-*r/ 71.0%

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

      2. neg-mul-1 71.0%

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

   7. Simplified 71.0%

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

   if -5.60000000000000031e-10 < d < 3.39999999999999974e-7

   1. Initial program 72.0%

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

      1. fma-neg 72.0%

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

      2. distribute-rgt-neg-out 72.0%

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

      3. +-commutative 72.0%

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

      4. fma-define 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in c around inf 84.6%

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

      1. mul-1-neg 84.6%

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

      2. unsub-neg 84.6%

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

      3. associate-/l* 83.5%

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

   7. Simplified 83.5%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.6 \cdot 10^{-10} \lor \neg \left(d \leq 3.4 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-12} \lor \neg \left(d \leq 2.3 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.1e-12) (not (<= d 2.3e-68))) (/ a (- d)) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.1e-12) || !(d <= 2.3e-68)) {
		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 <= (-1.1d-12)) .or. (.not. (d <= 2.3d-68))) 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 <= -1.1e-12) || !(d <= 2.3e-68)) {
		tmp = a / -d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.1e-12) or not (d <= 2.3e-68):
		tmp = a / -d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.1e-12) || !(d <= 2.3e-68))
		tmp = Float64(a / Float64(-d));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.1e-12) || ~((d <= 2.3e-68)))
		tmp = a / -d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.1e-12], N[Not[LessEqual[d, 2.3e-68]], $MachinePrecision]], N[(a / (-d)), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.09999999999999996e-12 or 2.29999999999999997e-68 < d

   1. Initial program 55.9%

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

      1. fma-neg 55.9%

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

      2. distribute-rgt-neg-out 55.9%

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

      3. +-commutative 55.9%

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

      4. fma-define 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in c around 0 67.4%

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

      1. associate-*r/ 67.4%

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

      2. neg-mul-1 67.4%

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

   7. Simplified 67.4%

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

   if -1.09999999999999996e-12 < d < 2.29999999999999997e-68

   1. Initial program 71.0%

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

      1. fma-neg 71.0%

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

      2. distribute-rgt-neg-out 71.0%

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

      3. +-commutative 71.0%

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

      4. fma-define 71.0%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in c around inf 64.4%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-12} \lor \neg \left(d \leq 2.3 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8 \cdot 10^{+104} \lor \neg \left(d \leq 1.55 \cdot 10^{+104}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -8e+104) (not (<= d 1.55e+104))) (/ a d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8e+104) || !(d <= 1.55e+104)) {
		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 <= (-8d+104)) .or. (.not. (d <= 1.55d+104))) 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 <= -8e+104) || !(d <= 1.55e+104)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -8e+104) or not (d <= 1.55e+104):
		tmp = a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -8e+104) || !(d <= 1.55e+104))
		tmp = Float64(a / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -8e+104) || ~((d <= 1.55e+104)))
		tmp = a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -8e+104], N[Not[LessEqual[d, 1.55e+104]], $MachinePrecision]], N[(a / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -8e104 or 1.55000000000000008e104 < d

   1. Initial program 38.5%

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

      1. fma-neg 38.5%

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

      2. distribute-rgt-neg-out 38.5%

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

      3. +-commutative 38.5%

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

      4. fma-define 38.4%

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

   3. Simplified 38.4%

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

   5. Taylor expanded in c around 0 80.6%

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

      1. associate-*r/ 80.6%

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

      2. neg-mul-1 80.6%

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

   7. Simplified 80.6%

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

      1. neg-sub0 80.6%

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

      2. sub-neg 80.6%

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

      3. add-sqr-sqrt 32.8%

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

      4. sqrt-unprod 36.5%

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

      5. sqr-neg 36.5%

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

      6. sqrt-unprod 13.9%

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

      7. add-sqr-sqrt 25.2%

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

   9. Applied egg-rr 25.2%

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

       1. +-lft-identity 25.2%

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

   11. Simplified 25.2%

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

   if -8e104 < d < 1.55000000000000008e104

   1. Initial program 74.1%

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

      1. fma-neg 74.1%

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

      2. distribute-rgt-neg-out 74.1%

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

      3. +-commutative 74.1%

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

      4. fma-define 74.1%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in c around inf 50.6%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8 \cdot 10^{+104} \lor \neg \left(d \leq 1.55 \cdot 10^{+104}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 10.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

   1. fma-neg 62.6%

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

   2. distribute-rgt-neg-out 62.6%

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

   3. +-commutative 62.6%

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

   4. fma-define 62.6%

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

3. Simplified 62.6%

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

5. Taylor expanded in c around 0 44.7%

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

   1. associate-*r/ 44.7%

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

   2. neg-mul-1 44.7%

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

7. Simplified 44.7%

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

   1. neg-sub0 44.7%

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

   2. sub-neg 44.7%

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

   3. add-sqr-sqrt 17.2%

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

   4. sqrt-unprod 19.4%

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

   5. sqr-neg 19.4%

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

   6. sqrt-unprod 5.8%

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

   7. add-sqr-sqrt 10.6%

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

9. Applied egg-rr 10.6%

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

    1. +-lft-identity 10.6%

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

11. Simplified 10.6%

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

Developer Target 1: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024132 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d))))))

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