Complex division, imag part

Percentage Accurate: 61.0% → 91.5%

Time: 13.1s

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: 61.0% 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: 91.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{+145} \lor \neg \left(d \leq 9.6 \cdot 10^{+125}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\frac{\mathsf{hypot}\left(d, c\right)}{c}}, \frac{d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}} \cdot \left(-a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4.8e+145) (not (<= d 9.6e+125)))
   (fma (/ c (hypot c d)) (/ b (hypot c d)) (/ (- a) d))
   (fma
    (/ 1.0 (hypot c d))
    (/ b (/ (hypot d c) c))
    (* (/ d (pow (hypot c d) 2.0)) (- a)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.8e+145) || !(d <= 9.6e+125)) {
		tmp = fma((c / hypot(c, d)), (b / hypot(c, d)), (-a / d));
	} else {
		tmp = fma((1.0 / hypot(c, d)), (b / (hypot(d, c) / c)), ((d / pow(hypot(c, d), 2.0)) * -a));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4.8e+145) || !(d <= 9.6e+125))
		tmp = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(Float64(-a) / d));
	else
		tmp = fma(Float64(1.0 / hypot(c, d)), Float64(b / Float64(hypot(d, c) / c)), Float64(Float64(d / (hypot(c, d) ^ 2.0)) * Float64(-a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4.8e+145], N[Not[LessEqual[d, 9.6e+125]], $MachinePrecision]], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[((-a) / d), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[(N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] + N[(N[(d / N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.79999999999999984e145 or 9.5999999999999999e125 < d

   1. Initial program 29.8%

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

      1. div-sub 29.8%

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

      2. *-commutative 29.8%

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

      3. add-sqr-sqrt 29.8%

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

      4. times-frac 30.2%

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

      5. fma-neg 30.2%

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

      6. hypot-define 30.2%

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

      7. hypot-define 43.8%

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

      8. associate-/l* 47.9%

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

      9. add-sqr-sqrt 47.9%

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

      10. pow2 47.9%

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

      11. hypot-define 47.9%

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

   4. Applied egg-rr 47.9%

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

   5. Taylor expanded in d around inf 97.4%

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

   if -4.79999999999999984e145 < d < 9.5999999999999999e125

   1. Initial program 71.1%

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

      1. div-sub 67.8%

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

      2. *-un-lft-identity 67.8%

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

      3. add-sqr-sqrt 67.8%

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

      4. times-frac 67.9%

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

      5. fma-neg 68.4%

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

      6. hypot-define 68.4%

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

      7. hypot-define 73.5%

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

      8. associate-/l* 78.4%

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

      9. add-sqr-sqrt 78.4%

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

      10. pow2 78.4%

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

      11. hypot-define 78.4%

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

   4. Applied egg-rr 78.4%

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

      1. div-inv 78.3%

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

      2. *-commutative 78.3%

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

      3. associate-*l* 89.3%

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

      4. div-inv 89.4%

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

      5. hypot-undefine 75.1%

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

      6. +-commutative 75.1%

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

      7. hypot-define 89.4%

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

   6. Applied egg-rr 89.4%

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

      1. *-commutative 89.4%

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

      2. associate-/r/ 90.4%

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

   8. Simplified 90.4%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{+145} \lor \neg \left(d \leq 9.6 \cdot 10^{+125}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\frac{\mathsf{hypot}\left(d, c\right)}{c}}, \frac{d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}} \cdot \left(-a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{c}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;d \leq -1.2 \cdot 10^{+135} \lor \neg \left(d \leq 9.2 \cdot 10^{+125}\right):\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_0, \frac{-a}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_0, \frac{d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}} \cdot \left(-a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ b (hypot c d))) (t_1 (/ c (hypot c d))))
   (if (or (<= d -1.2e+135) (not (<= d 9.2e+125)))
     (fma t_1 t_0 (/ (- a) d))
     (fma t_1 t_0 (* (/ d (pow (hypot c d) 2.0)) (- a))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = b / hypot(c, d);
	double t_1 = c / hypot(c, d);
	double tmp;
	if ((d <= -1.2e+135) || !(d <= 9.2e+125)) {
		tmp = fma(t_1, t_0, (-a / d));
	} else {
		tmp = fma(t_1, t_0, ((d / pow(hypot(c, d), 2.0)) * -a));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(b / hypot(c, d))
	t_1 = Float64(c / hypot(c, d))
	tmp = 0.0
	if ((d <= -1.2e+135) || !(d <= 9.2e+125))
		tmp = fma(t_1, t_0, Float64(Float64(-a) / d));
	else
		tmp = fma(t_1, t_0, Float64(Float64(d / (hypot(c, d) ^ 2.0)) * Float64(-a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[d, -1.2e+135], N[Not[LessEqual[d, 9.2e+125]], $MachinePrecision]], N[(t$95$1 * t$95$0 + N[((-a) / d), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * t$95$0 + N[(N[(d / N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.19999999999999999e135 or 9.20000000000000051e125 < d

   1. Initial program 29.7%

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

      1. div-sub 29.7%

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

      2. *-commutative 29.7%

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

      3. add-sqr-sqrt 29.7%

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

      4. times-frac 30.1%

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

      5. fma-neg 30.1%

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

      6. hypot-define 30.1%

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

      7. hypot-define 43.0%

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

      8. associate-/l* 49.4%

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

      9. add-sqr-sqrt 49.4%

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

      10. pow2 49.4%

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

      11. hypot-define 49.4%

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

   4. Applied egg-rr 49.4%

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

   5. Taylor expanded in d around inf 96.2%

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

   if -1.19999999999999999e135 < d < 9.20000000000000051e125

   1. Initial program 72.1%

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

      1. div-sub 68.7%

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

      2. *-commutative 68.7%

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

      3. add-sqr-sqrt 68.7%

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

      4. times-frac 70.8%

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

      5. fma-neg 70.8%

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

      6. hypot-define 70.8%

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

      7. hypot-define 85.3%

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

      8. associate-/l* 89.3%

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

      9. add-sqr-sqrt 89.3%

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

      10. pow2 89.3%

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

      11. hypot-define 89.3%

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

   4. Applied egg-rr 89.3%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.2 \cdot 10^{+135} \lor \neg \left(d \leq 9.2 \cdot 10^{+125}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}} \cdot \left(-a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \mathbf{if}\;d \leq -5.2 \cdot 10^{-23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-162}:\\ \;\;\;\;\frac{d \cdot a}{c} \cdot \frac{-1}{c} + \frac{b}{c}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+51}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{+67}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma (/ c (hypot c d)) (/ b (hypot c d)) (/ (- a) d))))
   (if (<= d -5.2e-23)
     t_0
     (if (<= d 3.3e-162)
       (+ (* (/ (* d a) c) (/ -1.0 c)) (/ b c))
       (if (<= d 1.7e+51)
         (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))
         (if (<= d 1.8e+67) (- (/ b c) (* (/ d c) (/ a c))) 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 <= -5.2e-23) {
		tmp = t_0;
	} else if (d <= 3.3e-162) {
		tmp = (((d * a) / c) * (-1.0 / c)) + (b / c);
	} else if (d <= 1.7e+51) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else if (d <= 1.8e+67) {
		tmp = (b / c) - ((d / c) * (a / c));
	} 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(Float64(-a) / d))
	tmp = 0.0
	if (d <= -5.2e-23)
		tmp = t_0;
	elseif (d <= 3.3e-162)
		tmp = Float64(Float64(Float64(Float64(d * a) / c) * Float64(-1.0 / c)) + Float64(b / c));
	elseif (d <= 1.7e+51)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 1.8e+67)
		tmp = Float64(Float64(b / c) - Float64(Float64(d / c) * Float64(a / c)));
	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, -5.2e-23], t$95$0, If[LessEqual[d, 3.3e-162], N[(N[(N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision]), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.7e+51], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.8e+67], N[(N[(b / c), $MachinePrecision] - N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.2e-23 or 1.7999999999999999e67 < d

   1. Initial program 42.2%

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

      1. div-sub 42.2%

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

      2. *-commutative 42.2%

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

      3. add-sqr-sqrt 42.2%

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

      4. times-frac 41.7%

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

      5. fma-neg 41.7%

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

      6. hypot-define 41.7%

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

      7. hypot-define 53.2%

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

      8. associate-/l* 63.7%

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

      9. add-sqr-sqrt 63.7%

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

      10. pow2 63.7%

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

      11. hypot-define 63.7%

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

   4. Applied egg-rr 63.7%

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

   5. Taylor expanded in d around inf 86.3%

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

   if -5.2e-23 < d < 3.30000000000000013e-162

   1. Initial program 72.7%

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

   3. Taylor expanded in c around inf 85.7%

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

      1. *-un-lft-identity 85.7%

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

      2. pow2 85.7%

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

      3. times-frac 91.0%

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

      4. *-commutative 91.0%

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

   5. Applied egg-rr 91.0%

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

   if 3.30000000000000013e-162 < d < 1.69999999999999992e51

   1. Initial program 82.6%

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

   if 1.69999999999999992e51 < d < 1.7999999999999999e67

   1. Initial program 26.8%

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

   3. Taylor expanded in c around inf 76.8%

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

      1. *-commutative 76.8%

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

      2. pow2 76.8%

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

      3. times-frac 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{d}{c} \cdot \frac{a}{c}\right)} + \frac{b}{c} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-162}:\\ \;\;\;\;\frac{d \cdot a}{c} \cdot \frac{-1}{c} + \frac{b}{c}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+51}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{+67}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.65 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(c, d\right)}, \frac{c \cdot b}{-d}, \frac{-a}{d}\right)\\ \mathbf{elif}\;d \leq 1.95 \cdot 10^{-163}:\\ \;\;\;\;\frac{d \cdot a}{c} \cdot \frac{-1}{c} + \frac{b}{c}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+51}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{+67}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{b}{{d}^{2}} - \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.65e-22)
   (fma (/ 1.0 (hypot c d)) (/ (* c b) (- d)) (/ (- a) d))
   (if (<= d 1.95e-163)
     (+ (* (/ (* d a) c) (/ -1.0 c)) (/ b c))
     (if (<= d 1.7e+51)
       (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))
       (if (<= d 1.8e+67)
         (- (/ b c) (* (/ d c) (/ a c)))
         (- (* c (/ b (pow d 2.0))) (/ a d)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.65e-22) {
		tmp = fma((1.0 / hypot(c, d)), ((c * b) / -d), (-a / d));
	} else if (d <= 1.95e-163) {
		tmp = (((d * a) / c) * (-1.0 / c)) + (b / c);
	} else if (d <= 1.7e+51) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else if (d <= 1.8e+67) {
		tmp = (b / c) - ((d / c) * (a / c));
	} else {
		tmp = (c * (b / pow(d, 2.0))) - (a / d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.65e-22)
		tmp = fma(Float64(1.0 / hypot(c, d)), Float64(Float64(c * b) / Float64(-d)), Float64(Float64(-a) / d));
	elseif (d <= 1.95e-163)
		tmp = Float64(Float64(Float64(Float64(d * a) / c) * Float64(-1.0 / c)) + Float64(b / c));
	elseif (d <= 1.7e+51)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 1.8e+67)
		tmp = Float64(Float64(b / c) - Float64(Float64(d / c) * Float64(a / c)));
	else
		tmp = Float64(Float64(c * Float64(b / (d ^ 2.0))) - Float64(a / d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.65e-22], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(c * b), $MachinePrecision] / (-d)), $MachinePrecision] + N[((-a) / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.95e-163], N[(N[(N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision]), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.7e+51], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.8e+67], N[(N[(b / c), $MachinePrecision] - N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(b / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.65e-22

   1. Initial program 47.1%

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

      1. div-sub 47.1%

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

      2. *-un-lft-identity 47.1%

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

      3. add-sqr-sqrt 47.1%

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

      4. times-frac 47.1%

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

      5. fma-neg 47.1%

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

      6. hypot-define 47.1%

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

      7. hypot-define 53.0%

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

      8. associate-/l* 65.9%

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

      9. add-sqr-sqrt 65.9%

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

      10. pow2 65.9%

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

      11. hypot-define 65.9%

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

   4. Applied egg-rr 65.9%

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

      1. div-inv 65.9%

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

      2. *-commutative 65.9%

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

      3. associate-*l* 71.5%

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

      4. div-inv 71.5%

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

      5. hypot-undefine 60.3%

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

      6. +-commutative 60.3%

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

      7. hypot-define 71.5%

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

   6. Applied egg-rr 71.5%

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

      1. *-commutative 71.5%

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

      2. associate-/r/ 70.2%

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

   8. Simplified 70.2%

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

   9. Taylor expanded in d around inf 83.5%

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

   10. Taylor expanded in d around -inf 79.4%

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

       1. associate-*r/ 79.4%

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

       2. neg-mul-1 79.4%

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

       3. distribute-rgt-neg-in 79.4%

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

   12. Simplified 79.4%

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

   if -1.65e-22 < d < 1.9500000000000001e-163

   1. Initial program 72.7%

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

   3. Taylor expanded in c around inf 85.7%

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

      1. *-un-lft-identity 85.7%

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

      2. pow2 85.7%

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

      3. times-frac 91.0%

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

      4. *-commutative 91.0%

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

   5. Applied egg-rr 91.0%

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

   if 1.9500000000000001e-163 < d < 1.69999999999999992e51

   1. Initial program 82.6%

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

   if 1.69999999999999992e51 < d < 1.7999999999999999e67

   1. Initial program 26.8%

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

   3. Taylor expanded in c around inf 76.8%

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

      1. *-commutative 76.8%

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

      2. pow2 76.8%

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

      3. times-frac 100.0%

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

   5. Applied egg-rr 100.0%

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

   if 1.7999999999999999e67 < d

   1. Initial program 35.0%

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

   3. Taylor expanded in c around 0 75.4%

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

      1. +-commutative 75.4%

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

      2. mul-1-neg 75.4%

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

      3. unsub-neg 75.4%

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

      4. *-commutative 75.4%

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

      5. associate-/l* 80.3%

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

   5. Simplified 80.3%

      \[\leadsto \color{blue}{c \cdot \frac{b}{{d}^{2}} - \frac{a}{d}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.65 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(c, d\right)}, \frac{c \cdot b}{-d}, \frac{-a}{d}\right)\\ \mathbf{elif}\;d \leq 1.95 \cdot 10^{-163}:\\ \;\;\;\;\frac{d \cdot a}{c} \cdot \frac{-1}{c} + \frac{b}{c}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+51}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{+67}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{b}{{d}^{2}} - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \frac{b}{{d}^{2}} - \frac{a}{d}\\ \mathbf{if}\;d \leq -2.5 \cdot 10^{-22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-162}:\\ \;\;\;\;\frac{d \cdot a}{c} \cdot \frac{-1}{c} + \frac{b}{c}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+51}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+67}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c (/ b (pow d 2.0))) (/ a d))))
   (if (<= d -2.5e-22)
     t_0
     (if (<= d 3.3e-162)
       (+ (* (/ (* d a) c) (/ -1.0 c)) (/ b c))
       (if (<= d 1.7e+51)
         (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))
         (if (<= d 3e+67) (- (/ b c) (* (/ d c) (/ a c))) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * (b / pow(d, 2.0))) - (a / d);
	double tmp;
	if (d <= -2.5e-22) {
		tmp = t_0;
	} else if (d <= 3.3e-162) {
		tmp = (((d * a) / c) * (-1.0 / c)) + (b / c);
	} else if (d <= 1.7e+51) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else if (d <= 3e+67) {
		tmp = (b / c) - ((d / c) * (a / c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c * (b / (d ** 2.0d0))) - (a / d)
    if (d <= (-2.5d-22)) then
        tmp = t_0
    else if (d <= 3.3d-162) then
        tmp = (((d * a) / c) * ((-1.0d0) / c)) + (b / c)
    else if (d <= 1.7d+51) then
        tmp = ((c * b) - (d * a)) / ((c * c) + (d * d))
    else if (d <= 3d+67) then
        tmp = (b / c) - ((d / c) * (a / c))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (c * (b / Math.pow(d, 2.0))) - (a / d);
	double tmp;
	if (d <= -2.5e-22) {
		tmp = t_0;
	} else if (d <= 3.3e-162) {
		tmp = (((d * a) / c) * (-1.0 / c)) + (b / c);
	} else if (d <= 1.7e+51) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else if (d <= 3e+67) {
		tmp = (b / c) - ((d / c) * (a / c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (c * (b / math.pow(d, 2.0))) - (a / d)
	tmp = 0
	if d <= -2.5e-22:
		tmp = t_0
	elif d <= 3.3e-162:
		tmp = (((d * a) / c) * (-1.0 / c)) + (b / c)
	elif d <= 1.7e+51:
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d))
	elif d <= 3e+67:
		tmp = (b / c) - ((d / c) * (a / c))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * Float64(b / (d ^ 2.0))) - Float64(a / d))
	tmp = 0.0
	if (d <= -2.5e-22)
		tmp = t_0;
	elseif (d <= 3.3e-162)
		tmp = Float64(Float64(Float64(Float64(d * a) / c) * Float64(-1.0 / c)) + Float64(b / c));
	elseif (d <= 1.7e+51)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 3e+67)
		tmp = Float64(Float64(b / c) - Float64(Float64(d / c) * Float64(a / c)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (c * (b / (d ^ 2.0))) - (a / d);
	tmp = 0.0;
	if (d <= -2.5e-22)
		tmp = t_0;
	elseif (d <= 3.3e-162)
		tmp = (((d * a) / c) * (-1.0 / c)) + (b / c);
	elseif (d <= 1.7e+51)
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	elseif (d <= 3e+67)
		tmp = (b / c) - ((d / c) * (a / c));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * N[(b / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.5e-22], t$95$0, If[LessEqual[d, 3.3e-162], N[(N[(N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision]), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.7e+51], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3e+67], N[(N[(b / c), $MachinePrecision] - N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.49999999999999977e-22 or 3.0000000000000001e67 < d

   1. Initial program 42.2%

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

   3. Taylor expanded in c around 0 76.1%

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

      1. +-commutative 76.1%

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

      2. mul-1-neg 76.1%

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

      3. unsub-neg 76.1%

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

      4. *-commutative 76.1%

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

      5. associate-/l* 78.2%

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

   5. Simplified 78.2%

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

   if -2.49999999999999977e-22 < d < 3.30000000000000013e-162

   1. Initial program 72.7%

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

   3. Taylor expanded in c around inf 85.7%

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

      1. *-un-lft-identity 85.7%

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

      2. pow2 85.7%

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

      3. times-frac 91.0%

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

      4. *-commutative 91.0%

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

   5. Applied egg-rr 91.0%

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

   if 3.30000000000000013e-162 < d < 1.69999999999999992e51

   1. Initial program 82.6%

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

   if 1.69999999999999992e51 < d < 3.0000000000000001e67

   1. Initial program 26.8%

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

   3. Taylor expanded in c around inf 76.8%

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

      1. *-commutative 76.8%

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

      2. pow2 76.8%

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

      3. times-frac 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{d}{c} \cdot \frac{a}{c}\right)} + \frac{b}{c} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.5 \cdot 10^{-22}:\\ \;\;\;\;c \cdot \frac{b}{{d}^{2}} - \frac{a}{d}\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-162}:\\ \;\;\;\;\frac{d \cdot a}{c} \cdot \frac{-1}{c} + \frac{b}{c}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+51}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+67}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{b}{{d}^{2}} - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.1 \cdot 10^{-22}:\\ \;\;\;\;b \cdot \frac{c}{{d}^{2}} - \frac{a}{d}\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-162}:\\ \;\;\;\;\frac{d \cdot a}{c} \cdot \frac{-1}{c} + \frac{b}{c}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+51}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+67}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -3.1e-22)
   (- (* b (/ c (pow d 2.0))) (/ a d))
   (if (<= d 3.3e-162)
     (+ (* (/ (* d a) c) (/ -1.0 c)) (/ b c))
     (if (<= d 1.7e+51)
       (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))
       (if (<= d 2.8e+67) (- (/ b c) (* (/ d c) (/ a c))) (/ (- a) d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.1e-22) {
		tmp = (b * (c / pow(d, 2.0))) - (a / d);
	} else if (d <= 3.3e-162) {
		tmp = (((d * a) / c) * (-1.0 / c)) + (b / c);
	} else if (d <= 1.7e+51) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else if (d <= 2.8e+67) {
		tmp = (b / c) - ((d / c) * (a / c));
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-3.1d-22)) then
        tmp = (b * (c / (d ** 2.0d0))) - (a / d)
    else if (d <= 3.3d-162) then
        tmp = (((d * a) / c) * ((-1.0d0) / c)) + (b / c)
    else if (d <= 1.7d+51) then
        tmp = ((c * b) - (d * a)) / ((c * c) + (d * d))
    else if (d <= 2.8d+67) then
        tmp = (b / c) - ((d / c) * (a / c))
    else
        tmp = -a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.1e-22) {
		tmp = (b * (c / Math.pow(d, 2.0))) - (a / d);
	} else if (d <= 3.3e-162) {
		tmp = (((d * a) / c) * (-1.0 / c)) + (b / c);
	} else if (d <= 1.7e+51) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else if (d <= 2.8e+67) {
		tmp = (b / c) - ((d / c) * (a / c));
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -3.1e-22:
		tmp = (b * (c / math.pow(d, 2.0))) - (a / d)
	elif d <= 3.3e-162:
		tmp = (((d * a) / c) * (-1.0 / c)) + (b / c)
	elif d <= 1.7e+51:
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d))
	elif d <= 2.8e+67:
		tmp = (b / c) - ((d / c) * (a / c))
	else:
		tmp = -a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -3.1e-22)
		tmp = Float64(Float64(b * Float64(c / (d ^ 2.0))) - Float64(a / d));
	elseif (d <= 3.3e-162)
		tmp = Float64(Float64(Float64(Float64(d * a) / c) * Float64(-1.0 / c)) + Float64(b / c));
	elseif (d <= 1.7e+51)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 2.8e+67)
		tmp = Float64(Float64(b / c) - Float64(Float64(d / c) * Float64(a / c)));
	else
		tmp = Float64(Float64(-a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -3.1e-22)
		tmp = (b * (c / (d ^ 2.0))) - (a / d);
	elseif (d <= 3.3e-162)
		tmp = (((d * a) / c) * (-1.0 / c)) + (b / c);
	elseif (d <= 1.7e+51)
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	elseif (d <= 2.8e+67)
		tmp = (b / c) - ((d / c) * (a / c));
	else
		tmp = -a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -3.1e-22], N[(N[(b * N[(c / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.3e-162], N[(N[(N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision]), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.7e+51], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.8e+67], N[(N[(b / c), $MachinePrecision] - N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-a) / d), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -3.10000000000000013e-22

   1. Initial program 47.1%

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

      1. div-sub 47.1%

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

      2. *-un-lft-identity 47.1%

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

      3. add-sqr-sqrt 47.1%

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

      4. times-frac 47.1%

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

      5. fma-neg 47.1%

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

      6. hypot-define 47.1%

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

      7. hypot-define 53.0%

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

      8. associate-/l* 65.9%

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

      9. add-sqr-sqrt 65.9%

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

      10. pow2 65.9%

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

      11. hypot-define 65.9%

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

   4. Applied egg-rr 65.9%

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

      1. div-inv 65.9%

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

      2. *-commutative 65.9%

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

      3. associate-*l* 71.5%

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

      4. div-inv 71.5%

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

      5. hypot-undefine 60.3%

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

      6. +-commutative 60.3%

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

      7. hypot-define 71.5%

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

   6. Applied egg-rr 71.5%

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

      1. *-commutative 71.5%

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

      2. associate-/r/ 70.2%

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

   8. Simplified 70.2%

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

   9. Taylor expanded in c around 0 76.6%

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

       1. +-commutative 76.6%

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

       2. mul-1-neg 76.6%

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

       3. unsub-neg 76.6%

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

       4. associate-/l* 74.1%

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

   11. Simplified 74.1%

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

   if -3.10000000000000013e-22 < d < 3.30000000000000013e-162

   1. Initial program 72.7%

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

   3. Taylor expanded in c around inf 85.7%

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

      1. *-un-lft-identity 85.7%

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

      2. pow2 85.7%

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

      3. times-frac 91.0%

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

      4. *-commutative 91.0%

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

   5. Applied egg-rr 91.0%

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

   if 3.30000000000000013e-162 < d < 1.69999999999999992e51

   1. Initial program 82.6%

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

   if 1.69999999999999992e51 < d < 2.7999999999999998e67

   1. Initial program 26.8%

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

   3. Taylor expanded in c around inf 76.8%

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

      1. *-commutative 76.8%

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

      2. pow2 76.8%

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

      3. times-frac 100.0%

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

   5. Applied egg-rr 100.0%

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

   if 2.7999999999999998e67 < d

   1. Initial program 35.0%

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

   3. Taylor expanded in c around 0 78.3%

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

      1. associate-*r/ 78.3%

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

      2. neg-mul-1 78.3%

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

   5. Simplified 78.3%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.1 \cdot 10^{-22}:\\ \;\;\;\;b \cdot \frac{c}{{d}^{2}} - \frac{a}{d}\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-162}:\\ \;\;\;\;\frac{d \cdot a}{c} \cdot \frac{-1}{c} + \frac{b}{c}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+51}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+67}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ t_1 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -3.7 \cdot 10^{+93}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.25 \cdot 10^{-23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 3.25 \cdot 10^{-162}:\\ \;\;\;\;\frac{d \cdot a}{c} \cdot \frac{-1}{c} + \frac{b}{c}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)) (t_1 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (<= d -3.7e+93)
     t_0
     (if (<= d -1.25e-23)
       t_1
       (if (<= d 3.25e-162)
         (+ (* (/ (* d a) c) (/ -1.0 c)) (/ b c))
         (if (<= d 1.7e+51)
           t_1
           (if (<= d 2.2e+67) (- (/ b c) (* (/ d c) (/ a c))) t_0)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double t_1 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -3.7e+93) {
		tmp = t_0;
	} else if (d <= -1.25e-23) {
		tmp = t_1;
	} else if (d <= 3.25e-162) {
		tmp = (((d * a) / c) * (-1.0 / c)) + (b / c);
	} else if (d <= 1.7e+51) {
		tmp = t_1;
	} else if (d <= 2.2e+67) {
		tmp = (b / c) - ((d / c) * (a / c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -a / d
    t_1 = ((c * b) - (d * a)) / ((c * c) + (d * d))
    if (d <= (-3.7d+93)) then
        tmp = t_0
    else if (d <= (-1.25d-23)) then
        tmp = t_1
    else if (d <= 3.25d-162) then
        tmp = (((d * a) / c) * ((-1.0d0) / c)) + (b / c)
    else if (d <= 1.7d+51) then
        tmp = t_1
    else if (d <= 2.2d+67) then
        tmp = (b / c) - ((d / c) * (a / c))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double t_1 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -3.7e+93) {
		tmp = t_0;
	} else if (d <= -1.25e-23) {
		tmp = t_1;
	} else if (d <= 3.25e-162) {
		tmp = (((d * a) / c) * (-1.0 / c)) + (b / c);
	} else if (d <= 1.7e+51) {
		tmp = t_1;
	} else if (d <= 2.2e+67) {
		tmp = (b / c) - ((d / c) * (a / c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	t_1 = ((c * b) - (d * a)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -3.7e+93:
		tmp = t_0
	elif d <= -1.25e-23:
		tmp = t_1
	elif d <= 3.25e-162:
		tmp = (((d * a) / c) * (-1.0 / c)) + (b / c)
	elif d <= 1.7e+51:
		tmp = t_1
	elif d <= 2.2e+67:
		tmp = (b / c) - ((d / c) * (a / c))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	t_1 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -3.7e+93)
		tmp = t_0;
	elseif (d <= -1.25e-23)
		tmp = t_1;
	elseif (d <= 3.25e-162)
		tmp = Float64(Float64(Float64(Float64(d * a) / c) * Float64(-1.0 / c)) + Float64(b / c));
	elseif (d <= 1.7e+51)
		tmp = t_1;
	elseif (d <= 2.2e+67)
		tmp = Float64(Float64(b / c) - Float64(Float64(d / c) * Float64(a / c)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -a / d;
	t_1 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -3.7e+93)
		tmp = t_0;
	elseif (d <= -1.25e-23)
		tmp = t_1;
	elseif (d <= 3.25e-162)
		tmp = (((d * a) / c) * (-1.0 / c)) + (b / c);
	elseif (d <= 1.7e+51)
		tmp = t_1;
	elseif (d <= 2.2e+67)
		tmp = (b / c) - ((d / c) * (a / c));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.7e+93], t$95$0, If[LessEqual[d, -1.25e-23], t$95$1, If[LessEqual[d, 3.25e-162], N[(N[(N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision]), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.7e+51], t$95$1, If[LessEqual[d, 2.2e+67], N[(N[(b / c), $MachinePrecision] - N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.69999999999999987e93 or 2.2e67 < d

   1. Initial program 33.0%

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

   3. Taylor expanded in c around 0 76.7%

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

      1. associate-*r/ 76.7%

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

      2. neg-mul-1 76.7%

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

   5. Simplified 76.7%

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

   if -3.69999999999999987e93 < d < -1.2500000000000001e-23 or 3.24999999999999994e-162 < d < 1.69999999999999992e51

   1. Initial program 80.5%

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

   if -1.2500000000000001e-23 < d < 3.24999999999999994e-162

   1. Initial program 72.7%

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

   3. Taylor expanded in c around inf 85.7%

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

      1. *-un-lft-identity 85.7%

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

      2. pow2 85.7%

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

      3. times-frac 91.0%

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

      4. *-commutative 91.0%

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

   5. Applied egg-rr 91.0%

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

   if 1.69999999999999992e51 < d < 2.2e67

   1. Initial program 26.8%

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

   3. Taylor expanded in c around inf 76.8%

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

      1. *-commutative 76.8%

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

      2. pow2 76.8%

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

      3. times-frac 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{d}{c} \cdot \frac{a}{c}\right)} + \frac{b}{c} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.7 \cdot 10^{+93}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1.25 \cdot 10^{-23}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.25 \cdot 10^{-162}:\\ \;\;\;\;\frac{d \cdot a}{c} \cdot \frac{-1}{c} + \frac{b}{c}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+51}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-21} \lor \neg \left(d \leq 1.1 \cdot 10^{+70}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot a}{c} \cdot \frac{-1}{c} + \frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2e-21) (not (<= d 1.1e+70)))
   (/ (- a) d)
   (+ (* (/ (* d a) c) (/ -1.0 c)) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2e-21) || !(d <= 1.1e+70)) {
		tmp = -a / d;
	} else {
		tmp = (((d * a) / c) * (-1.0 / c)) + (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 <= (-2d-21)) .or. (.not. (d <= 1.1d+70))) then
        tmp = -a / d
    else
        tmp = (((d * a) / c) * ((-1.0d0) / c)) + (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 <= -2e-21) || !(d <= 1.1e+70)) {
		tmp = -a / d;
	} else {
		tmp = (((d * a) / c) * (-1.0 / c)) + (b / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2e-21) or not (d <= 1.1e+70):
		tmp = -a / d
	else:
		tmp = (((d * a) / c) * (-1.0 / c)) + (b / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2e-21) || !(d <= 1.1e+70))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(Float64(Float64(Float64(d * a) / c) * Float64(-1.0 / c)) + Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -2e-21) || ~((d <= 1.1e+70)))
		tmp = -a / d;
	else
		tmp = (((d * a) / c) * (-1.0 / c)) + (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2e-21], N[Not[LessEqual[d, 1.1e+70]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(N[(N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision]), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.99999999999999982e-21 or 1.1e70 < d

   1. Initial program 42.2%

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

   3. Taylor expanded in c around 0 71.9%

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

      1. associate-*r/ 71.9%

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

      2. neg-mul-1 71.9%

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

   5. Simplified 71.9%

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

   if -1.99999999999999982e-21 < d < 1.1e70

   1. Initial program 74.6%

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

   3. Taylor expanded in c around inf 79.0%

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

      1. *-un-lft-identity 79.0%

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

      2. pow2 79.0%

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

      3. times-frac 84.4%

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

      4. *-commutative 84.4%

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

   5. Applied egg-rr 84.4%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-21} \lor \neg \left(d \leq 1.1 \cdot 10^{+70}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot a}{c} \cdot \frac{-1}{c} + \frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-22} \lor \neg \left(d \leq 1.1 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.1e-22) (not (<= d 1.1e+68)))
   (/ (- a) d)
   (- (/ b c) (* (/ d c) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.1e-22) || !(d <= 1.1e+68)) {
		tmp = -a / d;
	} else {
		tmp = (b / c) - ((d / c) * (a / 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-22)) .or. (.not. (d <= 1.1d+68))) then
        tmp = -a / d
    else
        tmp = (b / c) - ((d / c) * (a / 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-22) || !(d <= 1.1e+68)) {
		tmp = -a / d;
	} else {
		tmp = (b / c) - ((d / c) * (a / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.1e-22) or not (d <= 1.1e+68):
		tmp = -a / d
	else:
		tmp = (b / c) - ((d / c) * (a / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.1e-22) || !(d <= 1.1e+68))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(d / c) * Float64(a / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.1e-22) || ~((d <= 1.1e+68)))
		tmp = -a / d;
	else
		tmp = (b / c) - ((d / c) * (a / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.1e-22], N[Not[LessEqual[d, 1.1e+68]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(N[(b / c), $MachinePrecision] - N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.1e-22 or 1.09999999999999994e68 < d

   1. Initial program 42.2%

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

   3. Taylor expanded in c around 0 71.9%

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

      1. associate-*r/ 71.9%

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

      2. neg-mul-1 71.9%

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

   5. Simplified 71.9%

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

   if -1.1e-22 < d < 1.09999999999999994e68

   1. Initial program 74.6%

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

   3. Taylor expanded in c around inf 79.0%

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

      1. *-commutative 79.0%

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

      2. pow2 79.0%

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

      3. times-frac 83.3%

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

   5. Applied egg-rr 83.3%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-22} \lor \neg \left(d \leq 1.1 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.2 \cdot 10^{-21} \lor \neg \left(d \leq 2.5 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.2e-21) (not (<= d 2.5e+67)))
   (/ (- a) d)
   (- (/ b c) (/ (* a (/ d c)) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.2e-21) || !(d <= 2.5e+67)) {
		tmp = -a / d;
	} else {
		tmp = (b / c) - ((a * (d / c)) / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-2.2d-21)) .or. (.not. (d <= 2.5d+67))) then
        tmp = -a / d
    else
        tmp = (b / c) - ((a * (d / c)) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.2e-21) || !(d <= 2.5e+67)) {
		tmp = -a / d;
	} else {
		tmp = (b / c) - ((a * (d / c)) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2.2e-21) or not (d <= 2.5e+67):
		tmp = -a / d
	else:
		tmp = (b / c) - ((a * (d / c)) / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.2e-21) || !(d <= 2.5e+67))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(a * Float64(d / c)) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -2.2e-21) || ~((d <= 2.5e+67)))
		tmp = -a / d;
	else
		tmp = (b / c) - ((a * (d / c)) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.2e-21], N[Not[LessEqual[d, 2.5e+67]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(N[(b / c), $MachinePrecision] - N[(N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.2000000000000001e-21 or 2.49999999999999988e67 < d

   1. Initial program 42.2%

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

   3. Taylor expanded in c around 0 71.9%

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

      1. associate-*r/ 71.9%

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

      2. neg-mul-1 71.9%

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

   5. Simplified 71.9%

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

   if -2.2000000000000001e-21 < d < 2.49999999999999988e67

   1. Initial program 74.6%

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

   3. Taylor expanded in c around inf 79.0%

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

      1. *-commutative 79.0%

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

      2. pow2 79.0%

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

      3. times-frac 83.3%

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

   5. Applied egg-rr 83.3%

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

      1. associate-*r/ 84.0%

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

   7. Applied egg-rr 84.0%

      \[\leadsto -1 \cdot \color{blue}{\frac{\frac{d}{c} \cdot a}{c}} + \frac{b}{c} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.2 \cdot 10^{-21} \lor \neg \left(d \leq 2.5 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{-22} \lor \neg \left(d \leq 1.8 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -5.8e-22) (not (<= d 1.8e+67))) (/ (- a) d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.8e-22) || !(d <= 1.8e+67)) {
		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 <= (-5.8d-22)) .or. (.not. (d <= 1.8d+67))) 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 <= -5.8e-22) || !(d <= 1.8e+67)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -5.8e-22) or not (d <= 1.8e+67):
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -5.8e-22) || !(d <= 1.8e+67))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -5.8e-22) || ~((d <= 1.8e+67)))
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -5.8e-22], N[Not[LessEqual[d, 1.8e+67]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -5.8000000000000003e-22 or 1.7999999999999999e67 < d

   1. Initial program 42.2%

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

   3. Taylor expanded in c around 0 71.9%

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

      1. associate-*r/ 71.9%

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

      2. neg-mul-1 71.9%

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

   5. Simplified 71.9%

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

   if -5.8000000000000003e-22 < d < 1.7999999999999999e67

   1. Initial program 74.6%

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

   3. Taylor expanded in c around inf 67.4%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{-22} \lor \neg \left(d \leq 1.8 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 6.5 \cdot 10^{+119}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (if (<= d 6.5e+119) (/ b c) (/ a d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 6.5e+119) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= 6.5d+119) then
        tmp = b / c
    else
        tmp = a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 6.5e+119) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= 6.5e+119:
		tmp = b / c
	else:
		tmp = a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= 6.5e+119)
		tmp = Float64(b / c);
	else
		tmp = Float64(a / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= 6.5e+119)
		tmp = b / c;
	else
		tmp = a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, 6.5e+119], N[(b / c), $MachinePrecision], N[(a / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 6.4999999999999997e119

   1. Initial program 64.7%

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

   3. Taylor expanded in c around inf 50.8%

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

   if 6.4999999999999997e119 < d

   1. Initial program 33.4%

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

      1. fma-neg 33.4%

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

      2. distribute-rgt-neg-out 33.4%

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

      3. *-un-lft-identity 33.4%

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

      4. add-sqr-sqrt 33.4%

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

      5. times-frac 33.3%

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

      6. hypot-define 33.3%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 7.5%

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

      9. sqr-neg 7.5%

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

      10. sqrt-prod 26.2%

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

      11. add-sqr-sqrt 26.2%

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

      12. hypot-define 28.6%

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

   4. Applied egg-rr 28.6%

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

   5. Taylor expanded in c around 0 24.7%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 6.5 \cdot 10^{+119}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 10.9% 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 59.7%

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

   1. fma-neg 59.7%

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

   2. distribute-rgt-neg-out 59.7%

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

   3. *-un-lft-identity 59.7%

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

   4. add-sqr-sqrt 59.7%

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

   5. times-frac 59.7%

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

   6. hypot-define 59.7%

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

   7. add-sqr-sqrt 29.5%

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

   8. sqrt-unprod 42.6%

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

   9. sqr-neg 42.6%

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

   10. sqrt-prod 20.2%

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

   11. add-sqr-sqrt 38.4%

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

   12. hypot-define 42.4%

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

4. Applied egg-rr 42.4%

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

5. Taylor expanded in c around 0 9.8%

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

6. Final simplification 9.8%

   \[\leadsto \frac{a}{d} \]
7. Add Preprocessing

Developer target: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024039 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

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

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