Complex division, real part

Percentage Accurate: 61.6% → 83.3%

Time: 5.4s

Alternatives: 10

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: 61.6% 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: 83.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{if}\;d \leq -2.6 \cdot 10^{+165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{d}{t\_0}, c \cdot \frac{a}{t\_0}\right)\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (/ (fma (/ c d) a b) d)))
   (if (<= d -2.6e+165)
     t_1
     (if (<= d -2.3e-60)
       (fma b (/ d t_0) (* c (/ a t_0)))
       (if (<= d 2.9e-126)
         (/ (fma (/ d c) b a) c)
         (if (<= d 5.5e+77) (/ (fma d b (* c a)) t_0) t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = fma((c / d), a, b) / d;
	double tmp;
	if (d <= -2.6e+165) {
		tmp = t_1;
	} else if (d <= -2.3e-60) {
		tmp = fma(b, (d / t_0), (c * (a / t_0)));
	} else if (d <= 2.9e-126) {
		tmp = fma((d / c), b, a) / c;
	} else if (d <= 5.5e+77) {
		tmp = fma(d, b, (c * a)) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = Float64(fma(Float64(c / d), a, b) / d)
	tmp = 0.0
	if (d <= -2.6e+165)
		tmp = t_1;
	elseif (d <= -2.3e-60)
		tmp = fma(b, Float64(d / t_0), Float64(c * Float64(a / t_0)));
	elseif (d <= 2.9e-126)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	elseif (d <= 5.5e+77)
		tmp = Float64(fma(d, b, Float64(c * a)) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2.6e+165], t$95$1, If[LessEqual[d, -2.3e-60], N[(b * N[(d / t$95$0), $MachinePrecision] + N[(c * N[(a / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.9e-126], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 5.5e+77], N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.6000000000000001e165 or 5.50000000000000036e77 < d

   1. Initial program 38.9%

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

   4. Applied rewrites 38.9%

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

   5. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 91.9

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

   7. Applied rewrites 91.9%

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

   if -2.6000000000000001e165 < d < -2.3000000000000001e-60

   1. Initial program 68.8%

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

   4. Applied rewrites 83.2%

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

   if -2.3000000000000001e-60 < d < 2.89999999999999988e-126

   1. Initial program 65.8%

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

   4. Applied rewrites 65.8%

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

   5. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 93.8

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

   7. Applied rewrites 93.8%

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

   if 2.89999999999999988e-126 < d < 5.50000000000000036e77

   1. Initial program 87.7%

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

   4. Applied rewrites 87.7%

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

Alternative 2: 64.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{+81}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -2.35 \cdot 10^{-63}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot d\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-204}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.55e+81)
   (/ b d)
   (if (<= d -2.35e-63)
     (* (/ b (fma d d (* c c))) d)
     (if (<= d 7.5e-204)
       (/ a c)
       (if (<= d 1.1e-5) (/ (fma a c (* b d)) (* c c)) (/ b d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.55e+81) {
		tmp = b / d;
	} else if (d <= -2.35e-63) {
		tmp = (b / fma(d, d, (c * c))) * d;
	} else if (d <= 7.5e-204) {
		tmp = a / c;
	} else if (d <= 1.1e-5) {
		tmp = fma(a, c, (b * d)) / (c * c);
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.55e+81)
		tmp = Float64(b / d);
	elseif (d <= -2.35e-63)
		tmp = Float64(Float64(b / fma(d, d, Float64(c * c))) * d);
	elseif (d <= 7.5e-204)
		tmp = Float64(a / c);
	elseif (d <= 1.1e-5)
		tmp = Float64(fma(a, c, Float64(b * d)) / Float64(c * c));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.55e+81], N[(b / d), $MachinePrecision], If[LessEqual[d, -2.35e-63], N[(N[(b / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision], If[LessEqual[d, 7.5e-204], N[(a / c), $MachinePrecision], If[LessEqual[d, 1.1e-5], N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.55e81 or 1.1e-5 < d

   1. Initial program 51.2%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if -1.55e81 < d < -2.35e-63

   1. Initial program 72.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 58.0

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

   5. Applied rewrites 58.0%

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

   if -2.35e-63 < d < 7.5000000000000003e-204

   1. Initial program 62.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 74.4

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

   5. Applied rewrites 74.4%

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

   if 7.5000000000000003e-204 < d < 1.1e-5

   1. Initial program 87.0%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.8

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

   5. Applied rewrites 70.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 70.8

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

   7. Applied rewrites 70.8%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{+81}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -2.35 \cdot 10^{-63}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot d\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-204}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{if}\;d \leq -7.5 \cdot 10^{+68}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ c d) a b) d)))
   (if (<= d -7.5e+68)
     t_0
     (if (<= d 2.9e-126)
       (/ (fma (/ d c) b a) c)
       (if (<= d 5.5e+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((c / d), a, b) / d;
	double tmp;
	if (d <= -7.5e+68) {
		tmp = t_0;
	} else if (d <= 2.9e-126) {
		tmp = fma((d / c), b, a) / c;
	} else if (d <= 5.5e+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(c / d), a, b) / d)
	tmp = 0.0
	if (d <= -7.5e+68)
		tmp = t_0;
	elseif (d <= 2.9e-126)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	elseif (d <= 5.5e+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[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -7.5e+68], t$95$0, If[LessEqual[d, 2.9e-126], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 5.5e+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 3 regimes

2. if d < -7.49999999999999959e68 or 5.50000000000000036e77 < d

   1. Initial program 45.5%

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

   4. Applied rewrites 45.5%

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

   5. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 89.3

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

   7. Applied rewrites 89.3%

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

   if -7.49999999999999959e68 < d < 2.89999999999999988e-126

   1. Initial program 66.7%

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

   4. Applied rewrites 66.7%

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

   5. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 86.0

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

   7. Applied rewrites 86.0%

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

   if 2.89999999999999988e-126 < d < 5.50000000000000036e77

   1. Initial program 87.7%

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

   4. Applied rewrites 87.7%

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

Alternative 4: 70.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{if}\;d \leq -9 \cdot 10^{-54}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-204}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ a d) c b) d)))
   (if (<= d -9e-54)
     t_0
     (if (<= d 7.5e-204)
       (/ a c)
       (if (<= d 9.5e-11) (/ (fma a c (* b d)) (* c c)) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((a / d), c, b) / d;
	double tmp;
	if (d <= -9e-54) {
		tmp = t_0;
	} else if (d <= 7.5e-204) {
		tmp = a / c;
	} else if (d <= 9.5e-11) {
		tmp = fma(a, c, (b * d)) / (c * c);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(a / d), c, b) / d)
	tmp = 0.0
	if (d <= -9e-54)
		tmp = t_0;
	elseif (d <= 7.5e-204)
		tmp = Float64(a / c);
	elseif (d <= 9.5e-11)
		tmp = Float64(fma(a, c, Float64(b * 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[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -9e-54], t$95$0, If[LessEqual[d, 7.5e-204], N[(a / c), $MachinePrecision], If[LessEqual[d, 9.5e-11], N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -8.9999999999999997e-54 or 9.49999999999999951e-11 < d

   1. Initial program 55.1%

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l/ N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-add N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 79.3

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

   5. Applied rewrites 79.3%

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

   if -8.9999999999999997e-54 < d < 7.5000000000000003e-204

   1. Initial program 63.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 73.7

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

   5. Applied rewrites 73.7%

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

   if 7.5000000000000003e-204 < d < 9.49999999999999951e-11

   1. Initial program 87.0%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.8

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

   5. Applied rewrites 70.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 70.8

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

   7. Applied rewrites 70.8%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{-54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-204}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7.5 \cdot 10^{+68} \lor \neg \left(d \leq 3000\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -7.5e+68) (not (<= d 3000.0)))
   (/ (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 <= -7.5e+68) || !(d <= 3000.0)) {
		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 <= -7.5e+68) || !(d <= 3000.0))
		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, -7.5e+68], N[Not[LessEqual[d, 3000.0]], $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 < -7.49999999999999959e68 or 3e3 < d

   1. Initial program 51.7%

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 87.0

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

   7. Applied rewrites 87.0%

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

   if -7.49999999999999959e68 < d < 3e3

   1. Initial program 71.8%

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 83.9

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

   7. Applied rewrites 83.9%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.5 \cdot 10^{+68} \lor \neg \left(d \leq 3000\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7.5 \cdot 10^{+68} \lor \neg \left(d \leq 3000\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -7.5e+68) (not (<= d 3000.0)))
   (/ (fma (/ c d) a b) d)
   (/ (fma (/ b c) d a) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7.5e+68) || !(d <= 3000.0)) {
		tmp = fma((c / d), a, b) / d;
	} else {
		tmp = fma((b / c), d, a) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -7.5e+68) || !(d <= 3000.0))
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	else
		tmp = Float64(fma(Float64(b / c), d, a) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -7.5e+68], N[Not[LessEqual[d, 3000.0]], $MachinePrecision]], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -7.49999999999999959e68 or 3e3 < d

   1. Initial program 51.7%

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 87.0

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

   7. Applied rewrites 87.0%

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

   if -7.49999999999999959e68 < d < 3e3

   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 inf

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

      1. lower-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 83.2

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

   5. Applied rewrites 83.2%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.5 \cdot 10^{+68} \lor \neg \left(d \leq 3000\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7.5 \cdot 10^{+68} \lor \neg \left(d \leq 3000\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -7.5e+68) (not (<= d 3000.0)))
   (/ (fma (/ a d) c b) d)
   (/ (fma (/ b c) d a) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7.5e+68) || !(d <= 3000.0)) {
		tmp = fma((a / d), c, b) / d;
	} else {
		tmp = fma((b / c), d, a) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -7.5e+68) || !(d <= 3000.0))
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	else
		tmp = Float64(fma(Float64(b / c), d, a) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -7.5e+68], N[Not[LessEqual[d, 3000.0]], $MachinePrecision]], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -7.49999999999999959e68 or 3e3 < d

   1. Initial program 51.7%

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l/ N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-add N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if -7.49999999999999959e68 < d < 3e3

   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 inf

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

      1. lower-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 83.2

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

   5. Applied rewrites 83.2%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.5 \cdot 10^{+68} \lor \neg \left(d \leq 3000\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{+81}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -2.35 \cdot 10^{-63}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot d\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.55e+81)
   (/ b d)
   (if (<= d -2.35e-63)
     (* (/ b (fma d d (* c c))) d)
     (if (<= d 1.1e-5) (/ a c) (/ b d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.55e+81) {
		tmp = b / d;
	} else if (d <= -2.35e-63) {
		tmp = (b / fma(d, d, (c * c))) * d;
	} else if (d <= 1.1e-5) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.55e+81)
		tmp = Float64(b / d);
	elseif (d <= -2.35e-63)
		tmp = Float64(Float64(b / fma(d, d, Float64(c * c))) * d);
	elseif (d <= 1.1e-5)
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.55e+81], N[(b / d), $MachinePrecision], If[LessEqual[d, -2.35e-63], N[(N[(b / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision], If[LessEqual[d, 1.1e-5], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.55e81 or 1.1e-5 < d

   1. Initial program 51.2%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if -1.55e81 < d < -2.35e-63

   1. Initial program 72.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 58.0

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

   5. Applied rewrites 58.0%

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

   if -2.35e-63 < d < 1.1e-5

   1. Initial program 72.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 69.4

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

   5. Applied rewrites 69.4%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{+81}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -2.35 \cdot 10^{-63}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot d\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{-50} \lor \neg \left(d \leq 1.1 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.25e-50) (not (<= d 1.1e-5))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.25e-50) || !(d <= 1.1e-5)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	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 ((d <= (-1.25d-50)) .or. (.not. (d <= 1.1d-5))) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.25e-50) || !(d <= 1.1e-5)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.25e-50) or not (d <= 1.1e-5):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.25e-50) || !(d <= 1.1e-5))
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.25e-50) || ~((d <= 1.1e-5)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.25e-50], N[Not[LessEqual[d, 1.1e-5]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.24999999999999992e-50 or 1.1e-5 < d

   1. Initial program 55.1%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 71.4

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

   5. Applied rewrites 71.4%

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

   if -1.24999999999999992e-50 < d < 1.1e-5

   1. Initial program 72.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 69.1

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

   5. Applied rewrites 69.1%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{-50} \lor \neg \left(d \leq 1.1 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.2% 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 63.2%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 42.3

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

5. Applied rewrites 42.3%

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

6. Final simplification 42.3%

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

Developer Target 1: 99.1% 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 2024360 
(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))))