Complex division, real part

Percentage Accurate: 60.7% → 81.9%

Time: 5.1s

Alternatives: 9

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{if}\;c \leq -2.4 \cdot 10^{+91}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -2.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 7 \cdot 10^{-118}:\\ \;\;\;\;\frac{b - \frac{\mathsf{fma}\left(-a, c, \frac{\left(b \cdot c\right) \cdot c}{d}\right)}{d}}{d}\\ \mathbf{elif}\;c \leq 2.25 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ d c) b a) c)))
   (if (<= c -2.4e+91)
     t_0
     (if (<= c -2.2e-83)
       (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
       (if (<= c 7e-118)
         (/ (- b (/ (fma (- a) c (/ (* (* b c) c) d)) d)) d)
         (if (<= c 2.25e+77) (/ (fma d b (* c a)) (fma d d (* c c))) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((d / c), b, a) / c;
	double tmp;
	if (c <= -2.4e+91) {
		tmp = t_0;
	} else if (c <= -2.2e-83) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (c <= 7e-118) {
		tmp = (b - (fma(-a, c, (((b * c) * c) / d)) / d)) / d;
	} else if (c <= 2.25e+77) {
		tmp = fma(d, b, (c * a)) / fma(d, d, (c * c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(d / c), b, a) / c)
	tmp = 0.0
	if (c <= -2.4e+91)
		tmp = t_0;
	elseif (c <= -2.2e-83)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 7e-118)
		tmp = Float64(Float64(b - Float64(fma(Float64(-a), c, Float64(Float64(Float64(b * c) * c) / d)) / d)) / d);
	elseif (c <= 2.25e+77)
		tmp = Float64(fma(d, b, Float64(c * a)) / fma(d, d, Float64(c * c)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -2.4e+91], t$95$0, If[LessEqual[c, -2.2e-83], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 7e-118], N[(N[(b - N[(N[((-a) * c + N[(N[(N[(b * c), $MachinePrecision] * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.25e+77], N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.39999999999999983e91 or 2.25000000000000012e77 < c

   1. Initial program 39.7%

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

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 86.8%

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

   if -2.39999999999999983e91 < c < -2.20000000000000008e-83

   1. Initial program 81.7%

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

   if -2.20000000000000008e-83 < c < 7e-118

   1. Initial program 67.2%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \frac{b \cdot c}{{d}^{3}} + \frac{a}{{d}^{2}}\right) + \frac{b}{d}} \]

   5. Applied rewrites 89.3%

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

   if 7e-118 < c < 2.25000000000000012e77

   1. Initial program 84.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 84.1

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 84.1

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 84.1

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

   4. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 82.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{if}\;c \leq -2.4 \cdot 10^{+91}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -2.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{-117}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{elif}\;c \leq 2.25 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ d c) b a) c)))
   (if (<= c -2.4e+91)
     t_0
     (if (<= c -2.2e-83)
       (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
       (if (<= c 2.3e-117)
         (/ (fma (/ c d) a b) d)
         (if (<= c 2.25e+77) (/ (fma d b (* c a)) (fma d d (* c c))) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((d / c), b, a) / c;
	double tmp;
	if (c <= -2.4e+91) {
		tmp = t_0;
	} else if (c <= -2.2e-83) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (c <= 2.3e-117) {
		tmp = fma((c / d), a, b) / d;
	} else if (c <= 2.25e+77) {
		tmp = fma(d, b, (c * a)) / fma(d, d, (c * c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(d / c), b, a) / c)
	tmp = 0.0
	if (c <= -2.4e+91)
		tmp = t_0;
	elseif (c <= -2.2e-83)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 2.3e-117)
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	elseif (c <= 2.25e+77)
		tmp = Float64(fma(d, b, Float64(c * a)) / fma(d, d, Float64(c * c)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -2.4e+91], t$95$0, If[LessEqual[c, -2.2e-83], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.3e-117], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.25e+77], N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.39999999999999983e91 or 2.25000000000000012e77 < c

   1. Initial program 39.7%

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

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 86.8%

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

   if -2.39999999999999983e91 < c < -2.20000000000000008e-83

   1. Initial program 81.7%

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

   if -2.20000000000000008e-83 < c < 2.29999999999999994e-117

   1. Initial program 67.2%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 87.4%

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

   if 2.29999999999999994e-117 < c < 2.25000000000000012e77

   1. Initial program 84.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 84.1

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 84.1

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 84.1

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

   4. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 82.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ t_1 := \frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{if}\;c \leq -2.4 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -2.2 \cdot 10^{-83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{-117}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{elif}\;c \leq 2.25 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma d b (* c a)) (fma d d (* c c))))
        (t_1 (/ (fma (/ d c) b a) c)))
   (if (<= c -2.4e+91)
     t_1
     (if (<= c -2.2e-83)
       t_0
       (if (<= c 2.3e-117)
         (/ (fma (/ c d) a b) d)
         (if (<= c 2.25e+77) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, b, (c * a)) / fma(d, d, (c * c));
	double t_1 = fma((d / c), b, a) / c;
	double tmp;
	if (c <= -2.4e+91) {
		tmp = t_1;
	} else if (c <= -2.2e-83) {
		tmp = t_0;
	} else if (c <= 2.3e-117) {
		tmp = fma((c / d), a, b) / d;
	} else if (c <= 2.25e+77) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(d, b, Float64(c * a)) / fma(d, d, Float64(c * c)))
	t_1 = Float64(fma(Float64(d / c), b, a) / c)
	tmp = 0.0
	if (c <= -2.4e+91)
		tmp = t_1;
	elseif (c <= -2.2e-83)
		tmp = t_0;
	elseif (c <= 2.3e-117)
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	elseif (c <= 2.25e+77)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -2.4e+91], t$95$1, If[LessEqual[c, -2.2e-83], t$95$0, If[LessEqual[c, 2.3e-117], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.25e+77], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.39999999999999983e91 or 2.25000000000000012e77 < c

   1. Initial program 39.7%

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

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 86.8%

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

   if -2.39999999999999983e91 < c < -2.20000000000000008e-83 or 2.29999999999999994e-117 < c < 2.25000000000000012e77

   1. Initial program 83.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 83.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 83.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 83.0

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

   4. Applied rewrites 83.0%

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

   if -2.20000000000000008e-83 < c < 2.29999999999999994e-117

   1. Initial program 67.2%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 87.4%

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

Alternative 4: 72.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -9.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{+79}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a\\ \mathbf{elif}\;c \leq 2.65 \cdot 10^{+132}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a, b \cdot d\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -9.5e+64)
   (/ a c)
   (if (<= c 1.6e-17)
     (/ (fma (/ c d) a b) d)
     (if (<= c 3.8e+79)
       (* (/ c (fma d d (* c c))) a)
       (if (<= c 2.65e+132) (/ (fma c a (* b d)) (* c c)) (/ a c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -9.5e+64) {
		tmp = a / c;
	} else if (c <= 1.6e-17) {
		tmp = fma((c / d), a, b) / d;
	} else if (c <= 3.8e+79) {
		tmp = (c / fma(d, d, (c * c))) * a;
	} else if (c <= 2.65e+132) {
		tmp = fma(c, a, (b * d)) / (c * c);
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -9.5e+64)
		tmp = Float64(a / c);
	elseif (c <= 1.6e-17)
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	elseif (c <= 3.8e+79)
		tmp = Float64(Float64(c / fma(d, d, Float64(c * c))) * a);
	elseif (c <= 2.65e+132)
		tmp = Float64(fma(c, a, Float64(b * d)) / Float64(c * c));
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -9.5e+64], N[(a / c), $MachinePrecision], If[LessEqual[c, 1.6e-17], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 3.8e+79], N[(N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[c, 2.65e+132], N[(N[(c * a + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(a / c), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -9.50000000000000028e64 or 2.65e132 < c

   1. Initial program 37.9%

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

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 76.0%

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

   if -9.50000000000000028e64 < c < 1.6000000000000001e-17

   1. Initial program 71.8%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 80.0%

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

   if 1.6000000000000001e-17 < c < 3.8000000000000002e79

   1. Initial program 80.4%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 76.0%

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

   if 3.8000000000000002e79 < c < 2.65e132

   1. Initial program 84.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 84.9

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 84.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 84.9

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

   4. Applied rewrites 84.9%

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

   5. Taylor expanded in c around inf

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

   7. Applied rewrites 84.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 84.8

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

   9. Applied rewrites 84.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(c, a, b \cdot d\right)}}{c \cdot c} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 64.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 7 \cdot 10^{-117}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{+79}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a\\ \mathbf{elif}\;c \leq 2.65 \cdot 10^{+132}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a, b \cdot d\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.5e-14)
   (/ a c)
   (if (<= c 7e-117)
     (/ b d)
     (if (<= c 3.8e+79)
       (* (/ c (fma d d (* c c))) a)
       (if (<= c 2.65e+132) (/ (fma c a (* b d)) (* c c)) (/ a c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.5e-14) {
		tmp = a / c;
	} else if (c <= 7e-117) {
		tmp = b / d;
	} else if (c <= 3.8e+79) {
		tmp = (c / fma(d, d, (c * c))) * a;
	} else if (c <= 2.65e+132) {
		tmp = fma(c, a, (b * d)) / (c * c);
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.5e-14)
		tmp = Float64(a / c);
	elseif (c <= 7e-117)
		tmp = Float64(b / d);
	elseif (c <= 3.8e+79)
		tmp = Float64(Float64(c / fma(d, d, Float64(c * c))) * a);
	elseif (c <= 2.65e+132)
		tmp = Float64(fma(c, a, Float64(b * d)) / Float64(c * c));
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.5e-14], N[(a / c), $MachinePrecision], If[LessEqual[c, 7e-117], N[(b / d), $MachinePrecision], If[LessEqual[c, 3.8e+79], N[(N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[c, 2.65e+132], N[(N[(c * a + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(a / c), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -3.5000000000000002e-14 or 2.65e132 < c

   1. Initial program 41.8%

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

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 71.3%

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

   if -3.5000000000000002e-14 < c < 6.9999999999999997e-117

   1. Initial program 69.9%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 74.0%

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

   if 6.9999999999999997e-117 < c < 3.8000000000000002e79

   1. Initial program 82.0%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 71.1%

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

   if 3.8000000000000002e79 < c < 2.65e132

   1. Initial program 84.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 84.9

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 84.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 84.9

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

   4. Applied rewrites 84.9%

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

   5. Taylor expanded in c around inf

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

   7. Applied rewrites 84.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 84.8

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

   9. Applied rewrites 84.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(c, a, b \cdot d\right)}}{c \cdot c} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 64.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 7 \cdot 10^{-117}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{+79}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.5e-14)
   (/ a c)
   (if (<= c 7e-117)
     (/ b d)
     (if (<= c 7.5e+79) (* (/ c (fma d d (* c c))) a) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.5e-14) {
		tmp = a / c;
	} else if (c <= 7e-117) {
		tmp = b / d;
	} else if (c <= 7.5e+79) {
		tmp = (c / fma(d, d, (c * c))) * a;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.5e-14)
		tmp = Float64(a / c);
	elseif (c <= 7e-117)
		tmp = Float64(b / d);
	elseif (c <= 7.5e+79)
		tmp = Float64(Float64(c / fma(d, d, Float64(c * c))) * a);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.5e-14], N[(a / c), $MachinePrecision], If[LessEqual[c, 7e-117], N[(b / d), $MachinePrecision], If[LessEqual[c, 7.5e+79], N[(N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(a / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.5000000000000002e-14 or 7.49999999999999967e79 < c

   1. Initial program 46.6%

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

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 69.4%

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

   if -3.5000000000000002e-14 < c < 6.9999999999999997e-117

   1. Initial program 69.9%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 74.0%

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

   if 6.9999999999999997e-117 < c < 7.49999999999999967e79

   1. Initial program 82.0%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 78.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3 \cdot 10^{+39} \lor \neg \left(d \leq 4.9 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -3e+39) (not (<= d 4.9e-9)))
   (/ (fma (/ c d) a b) d)
   (/ (fma (/ d c) b a) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3e+39) || !(d <= 4.9e-9)) {
		tmp = fma((c / d), a, b) / d;
	} else {
		tmp = fma((d / c), b, a) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -3e+39) || !(d <= 4.9e-9))
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	else
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -3e+39], N[Not[LessEqual[d, 4.9e-9]], $MachinePrecision]], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -3e39 or 4.90000000000000004e-9 < d

   1. Initial program 48.6%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 76.1%

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

   if -3e39 < d < 4.90000000000000004e-9

   1. Initial program 73.6%

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

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 87.2%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3 \cdot 10^{+39} \lor \neg \left(d \leq 4.9 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.5 \cdot 10^{-14} \lor \neg \left(c \leq 8.5 \cdot 10^{-117}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -3.5e-14) (not (<= c 8.5e-117))) (/ a c) (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.5e-14) || !(c <= 8.5e-117)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-3.5d-14)) .or. (.not. (c <= 8.5d-117))) then
        tmp = a / c
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.5e-14) || !(c <= 8.5e-117)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -3.5e-14) or not (c <= 8.5e-117):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -3.5e-14) || !(c <= 8.5e-117))
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -3.5e-14) || ~((c <= 8.5e-117)))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -3.5e-14], N[Not[LessEqual[c, 8.5e-117]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.5000000000000002e-14 or 8.49999999999999981e-117 < c

   1. Initial program 55.2%

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

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 63.4%

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

   if -3.5000000000000002e-14 < c < 8.49999999999999981e-117

   1. Initial program 69.9%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 74.0%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.5 \cdot 10^{-14} \lor \neg \left(c \leq 8.5 \cdot 10^{-117}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.6% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

3. Taylor expanded in c around inf

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

5. Applied rewrites 44.2%

   \[\leadsto \color{blue}{\frac{a}{c}} \]
6. Add Preprocessing

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (< (fabs d) (fabs c))
   (/ (+ a (* b (/ d c))) (+ c (* d (/ d c))))
   (/ (+ b (* a (/ 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 = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    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 = (a + (b * (d / c))) / (c + (d * (d / c)))
    else
        tmp = (b + (a * (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 = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (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 = (a + (b * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (b + (a * (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(a + Float64(b * Float64(d / c))) / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(b + Float64(a * 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 = (a + (b * (d / c))) / (c + (d * (d / c)));
	else
		tmp = (b + (a * (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[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d + N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2025056 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

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

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