Complex division, imag part

Percentage Accurate: 61.8% → 98.8%

Time: 11.2s

Alternatives: 7

Speedup: 0.1×

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: 61.8% 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: 98.8% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(a, b, c, d):
	return ((c * (b / math.hypot(c, d))) - ((d / math.hypot(c, d)) * a)) / math.hypot(c, d)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

   1. div-sub 55.8%

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

   2. sub-neg 55.8%

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

   3. associate-/l* 58.1%

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

   4. fma-def 58.1%

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

   5. add-sqr-sqrt 58.1%

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

   6. pow2 58.1%

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

   7. fma-def 58.1%

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

   8. hypot-def 58.1%

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

   9. fma-def 58.1%

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

   10. add-sqr-sqrt 58.1%

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

   11. pow2 58.1%

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

   12. fma-def 58.1%

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

   13. hypot-def 58.1%

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

4. Applied egg-rr 58.1%

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

   1. sub-neg 58.1%

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

   2. associate-/r/ 54.7%

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

   3. associate-*l/ 55.8%

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

   4. *-commutative 55.8%

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

   5. *-commutative 55.8%

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

   6. *-lft-identity 55.8%

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

   7. times-frac 56.0%

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

   8. /-rgt-identity 56.0%

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

6. Simplified 56.0%

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

   1. unpow2 56.0%

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

   2. times-frac 75.3%

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

8. Applied egg-rr 75.3%

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

   1. *-commutative 75.3%

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

   2. associate-*l/ 77.2%

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

10. Simplified 77.2%

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

    1. associate-*r/ 76.0%

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

    2. *-commutative 76.0%

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

    3. associate-*r/ 74.2%

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

    4. associate-*l/ 74.5%

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

    5. unpow2 74.5%

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

    6. frac-times 95.4%

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

    7. associate-*r/ 96.9%

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

    8. sub-div 97.2%

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

12. Applied egg-rr 97.2%

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

13. Final simplification 97.2%

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

Alternative 2: 63.1% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(a, b, c, d):
	return ((b * (c / math.hypot(c, d))) / math.hypot(c, d)) - (a / d)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

   1. div-sub 55.8%

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

   2. sub-neg 55.8%

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

   3. associate-/l* 58.1%

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

   4. fma-def 58.1%

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

   5. add-sqr-sqrt 58.1%

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

   6. pow2 58.1%

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

   7. fma-def 58.1%

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

   8. hypot-def 58.1%

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

   9. fma-def 58.1%

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

   10. add-sqr-sqrt 58.1%

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

   11. pow2 58.1%

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

   12. fma-def 58.1%

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

   13. hypot-def 58.1%

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

4. Applied egg-rr 58.1%

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

   1. sub-neg 58.1%

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

   2. associate-/r/ 54.7%

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

   3. associate-*l/ 55.8%

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

   4. *-commutative 55.8%

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

   5. *-commutative 55.8%

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

   6. *-lft-identity 55.8%

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

   7. times-frac 56.0%

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

   8. /-rgt-identity 56.0%

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

6. Simplified 56.0%

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

   1. unpow2 56.0%

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

   2. times-frac 75.3%

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

8. Applied egg-rr 75.3%

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

   1. *-commutative 75.3%

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

   2. associate-*l/ 77.2%

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

10. Simplified 77.2%

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

11. Taylor expanded in d around inf 62.4%

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

12. Final simplification 62.4%

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

Alternative 3: 62.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(a, b, c, d):
	return (b - ((d / math.hypot(c, d)) * a)) / math.hypot(c, d)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

   1. div-sub 55.8%

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

   2. sub-neg 55.8%

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

   3. associate-/l* 58.1%

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

   4. fma-def 58.1%

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

   5. add-sqr-sqrt 58.1%

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

   6. pow2 58.1%

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

   7. fma-def 58.1%

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

   8. hypot-def 58.1%

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

   9. fma-def 58.1%

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

   10. add-sqr-sqrt 58.1%

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

   11. pow2 58.1%

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

   12. fma-def 58.1%

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

   13. hypot-def 58.1%

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

4. Applied egg-rr 58.1%

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

   1. sub-neg 58.1%

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

   2. associate-/r/ 54.7%

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

   3. associate-*l/ 55.8%

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

   4. *-commutative 55.8%

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

   5. *-commutative 55.8%

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

   6. *-lft-identity 55.8%

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

   7. times-frac 56.0%

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

   8. /-rgt-identity 56.0%

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

6. Simplified 56.0%

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

   1. unpow2 56.0%

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

   2. times-frac 75.3%

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

8. Applied egg-rr 75.3%

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

   1. *-commutative 75.3%

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

   2. associate-*l/ 77.2%

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

10. Simplified 77.2%

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

    1. associate-*r/ 76.0%

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

    2. *-commutative 76.0%

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

    3. associate-*r/ 74.2%

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

    4. associate-*l/ 74.5%

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

    5. unpow2 74.5%

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

    6. frac-times 95.4%

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

    7. associate-*r/ 96.9%

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

    8. sub-div 97.2%

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

12. Applied egg-rr 97.2%

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

13. Taylor expanded in c around inf 61.2%

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

14. Final simplification 61.2%

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

Alternative 4: 51.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{b}{c} - \frac{a}{c \cdot \left(c \cdot \frac{1}{d}\right)} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (- (/ b c) (/ a (* c (* c (/ 1.0 d))))))

C

double code(double a, double b, double c, double d) {
	return (b / c) - (a / (c * (c * (1.0 / 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 / (c * (c * (1.0d0 / d))))
end function

Java

public static double code(double a, double b, double c, double d) {
	return (b / c) - (a / (c * (c * (1.0 / d))));
}

Python

def code(a, b, c, d):
	return (b / c) - (a / (c * (c * (1.0 / d))))

Julia

function code(a, b, c, d)
	return Float64(Float64(b / c) - Float64(a / Float64(c * Float64(c * Float64(1.0 / d)))))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = (b / c) - (a / (c * (c * (1.0 / d))));
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(b / c), $MachinePrecision] - N[(a / N[(c * N[(c * N[(1.0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.2%

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

3. Taylor expanded in c around inf 50.7%

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

   1. +-commutative 50.7%

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

   2. mul-1-neg 50.7%

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

   3. unsub-neg 50.7%

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

   4. associate-/l* 52.5%

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

5. Simplified 52.5%

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

   1. div-inv 52.5%

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

   2. unpow2 52.5%

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

   3. associate-*l* 55.5%

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

7. Applied egg-rr 55.5%

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

8. Final simplification 55.5%

   \[\leadsto \frac{b}{c} - \frac{a}{c \cdot \left(c \cdot \frac{1}{d}\right)} \]
9. Add Preprocessing

Alternative 5: 51.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	return (b / c) - (a / (c * (c / 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 / (c * (c / d)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

3. Taylor expanded in c around inf 50.7%

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

   1. +-commutative 50.7%

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

   2. mul-1-neg 50.7%

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

   3. unsub-neg 50.7%

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

   4. associate-/l* 52.5%

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

5. Simplified 52.5%

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

   1. unpow2 52.5%

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

   2. *-un-lft-identity 52.5%

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

   3. times-frac 55.5%

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

7. Applied egg-rr 55.5%

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

8. Final simplification 55.5%

   \[\leadsto \frac{b}{c} - \frac{a}{c \cdot \frac{c}{d}} \]
9. Add Preprocessing

Alternative 6: 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 58.2%

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

3. Taylor expanded in b around 0 36.0%

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

   1. mul-1-neg 36.0%

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

   2. distribute-rgt-neg-in 36.0%

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

5. Simplified 36.0%

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

   1. *-commutative 36.0%

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

   2. fma-def 36.0%

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

   3. add-sqr-sqrt 36.0%

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

   4. fma-def 36.0%

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

   5. hypot-udef 36.0%

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

   6. fma-def 36.0%

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

   7. hypot-udef 36.0%

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

   8. times-frac 52.6%

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

   9. add-sqr-sqrt 26.4%

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

   10. sqrt-unprod 17.6%

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

   11. sqr-neg 17.6%

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

   12. sqrt-prod 7.1%

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

   13. add-sqr-sqrt 14.3%

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

7. Applied egg-rr 14.3%

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

8. Taylor expanded in d around inf 8.2%

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

9. Final simplification 8.2%

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

Alternative 7: 43.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

3. Taylor expanded in c around inf 45.8%

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

4. Final simplification 45.8%

   \[\leadsto \frac{b}{c} \]
5. Add Preprocessing

Developer target: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024033 
(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))))