Complex division, imag part

Percentage Accurate: 63.4% → 85.1%

Time: 5.0s

Alternatives: 8

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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 = ((b * c) - (a * d)) / ((c * c) + (d * d))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 63.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 = ((b * c) - (a * d)) / ((c * c) + (d * d))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \frac{\mathsf{fma}\left(\frac{c}{d}, b, a \cdot {\left(\frac{c}{d} \cdot -1\right)}^{2}\right) + -1 \cdot a}{d}\\ t_2 := \mathsf{fma}\left(a \cdot \frac{d}{t\_0}, -1, b \cdot \frac{c}{t\_0}\right)\\ \mathbf{if}\;d \leq -1.18 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -3.15 \cdot 10^{-96}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-140}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d \cdot a}{c}, -1, b\right)}{c}\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+175}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c)))
        (t_1
         (/ (+ (fma (/ c d) b (* a (pow (* (/ c d) -1.0) 2.0))) (* -1.0 a)) d))
        (t_2 (fma (* a (/ d t_0)) -1.0 (* b (/ c t_0)))))
   (if (<= d -1.18e+154)
     t_1
     (if (<= d -3.15e-96)
       t_2
       (if (<= d 2.7e-140)
         (/ (fma (/ (* d a) c) -1.0 b) c)
         (if (<= d 2.3e+175) t_2 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), b, (a * pow(((c / d) * -1.0), 2.0))) + (-1.0 * a)) / d;
	double t_2 = fma((a * (d / t_0)), -1.0, (b * (c / t_0)));
	double tmp;
	if (d <= -1.18e+154) {
		tmp = t_1;
	} else if (d <= -3.15e-96) {
		tmp = t_2;
	} else if (d <= 2.7e-140) {
		tmp = fma(((d * a) / c), -1.0, b) / c;
	} else if (d <= 2.3e+175) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = Float64(Float64(fma(Float64(c / d), b, Float64(a * (Float64(Float64(c / d) * -1.0) ^ 2.0))) + Float64(-1.0 * a)) / d)
	t_2 = fma(Float64(a * Float64(d / t_0)), -1.0, Float64(b * Float64(c / t_0)))
	tmp = 0.0
	if (d <= -1.18e+154)
		tmp = t_1;
	elseif (d <= -3.15e-96)
		tmp = t_2;
	elseif (d <= 2.7e-140)
		tmp = Float64(fma(Float64(Float64(d * a) / c), -1.0, b) / c);
	elseif (d <= 2.3e+175)
		tmp = t_2;
	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[(N[(c / d), $MachinePrecision] * b + N[(a * N[Power[N[(N[(c / d), $MachinePrecision] * -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * a), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * N[(d / t$95$0), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(b * N[(c / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.18e+154], t$95$1, If[LessEqual[d, -3.15e-96], t$95$2, If[LessEqual[d, 2.7e-140], N[(N[(N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision] * -1.0 + b), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.3e+175], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.18000000000000004e154 or 2.3e175 < d

   1. Initial program 32.7%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 83.7

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

   4. Applied rewrites 83.7%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 79.9

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

   7. Applied rewrites 79.9%

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

   8. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 89.4%

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

   if -1.18000000000000004e154 < d < -3.1499999999999998e-96 or 2.7e-140 < d < 2.3e175

   1. Initial program 73.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r* N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. div-add-rev N/A

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

      14. associate-*r/ N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 79.3%

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

   if -3.1499999999999998e-96 < d < 2.7e-140

   1. Initial program 71.9%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 90.7

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

   4. Applied rewrites 90.7%

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

Alternative 2: 82.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -5.7 \cdot 10^{+113}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d}\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-92}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-140}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d \cdot a}{c}, -1, b\right)}{c}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{+159}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, a \cdot {\left(\frac{c}{d} \cdot -1\right)}^{2}\right) + -1 \cdot a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))))
   (if (<= d -5.7e+113)
     (/ (fma b (/ c d) (* -1.0 a)) d)
     (if (<= d -1.15e-92)
       t_0
       (if (<= d 2.7e-140)
         (/ (fma (/ (* d a) c) -1.0 b) c)
         (if (<= d 2.7e+159)
           t_0
           (/
            (+ (fma (/ c d) b (* a (pow (* (/ c d) -1.0) 2.0))) (* -1.0 a))
            d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -5.7e+113) {
		tmp = fma(b, (c / d), (-1.0 * a)) / d;
	} else if (d <= -1.15e-92) {
		tmp = t_0;
	} else if (d <= 2.7e-140) {
		tmp = fma(((d * a) / c), -1.0, b) / c;
	} else if (d <= 2.7e+159) {
		tmp = t_0;
	} else {
		tmp = (fma((c / d), b, (a * pow(((c / d) * -1.0), 2.0))) + (-1.0 * a)) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -5.7e+113)
		tmp = Float64(fma(b, Float64(c / d), Float64(-1.0 * a)) / d);
	elseif (d <= -1.15e-92)
		tmp = t_0;
	elseif (d <= 2.7e-140)
		tmp = Float64(fma(Float64(Float64(d * a) / c), -1.0, b) / c);
	elseif (d <= 2.7e+159)
		tmp = t_0;
	else
		tmp = Float64(Float64(fma(Float64(c / d), b, Float64(a * (Float64(Float64(c / d) * -1.0) ^ 2.0))) + Float64(-1.0 * a)) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.7e+113], N[(N[(b * N[(c / d), $MachinePrecision] + N[(-1.0 * a), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -1.15e-92], t$95$0, If[LessEqual[d, 2.7e-140], N[(N[(N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision] * -1.0 + b), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.7e+159], t$95$0, N[(N[(N[(N[(c / d), $MachinePrecision] * b + N[(a * N[Power[N[(N[(c / d), $MachinePrecision] * -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * a), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.6999999999999998e113

   1. Initial program 39.5%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 81.4

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

   4. Applied rewrites 81.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 86.3

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

   6. Applied rewrites 86.3%

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

   if -5.6999999999999998e113 < d < -1.15000000000000008e-92 or 2.7e-140 < d < 2.70000000000000008e159

   1. Initial program 74.6%

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

   if -1.15000000000000008e-92 < d < 2.7e-140

   1. Initial program 72.0%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 90.6

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

   4. Applied rewrites 90.6%

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

   if 2.70000000000000008e159 < d

   1. Initial program 32.8%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 83.1

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

   4. Applied rewrites 83.1%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 79.1

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

   7. Applied rewrites 79.1%

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

   8. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 88.4%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{c}{d}, b, a \cdot {\left(\frac{c}{d} \cdot -1\right)}^{2}\right) + -1 \cdot a}{d}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 77.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(a \cdot \frac{d}{c}, -1, b\right)}{c}\\ \mathbf{if}\;c \leq -350000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 7 \cdot 10^{-41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (* a (/ d c)) -1.0 b) c)))
   (if (<= c -350000000.0)
     t_0
     (if (<= c 7e-41) (/ (fma b (/ c d) (* -1.0 a)) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((a * (d / c)), -1.0, b) / c;
	double tmp;
	if (c <= -350000000.0) {
		tmp = t_0;
	} else if (c <= 7e-41) {
		tmp = fma(b, (c / d), (-1.0 * a)) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(a * Float64(d / c)), -1.0, b) / c)
	tmp = 0.0
	if (c <= -350000000.0)
		tmp = t_0;
	elseif (c <= 7e-41)
		tmp = Float64(fma(b, Float64(c / d), Float64(-1.0 * a)) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision] * -1.0 + b), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -350000000.0], t$95$0, If[LessEqual[c, 7e-41], N[(N[(b * N[(c / d), $MachinePrecision] + N[(-1.0 * a), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -3.5e8 or 6.9999999999999999e-41 < c

   1. Initial program 53.3%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 70.6

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

   4. Applied rewrites 70.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.0

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

   6. Applied rewrites 74.0%

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

   if -3.5e8 < c < 6.9999999999999999e-41

   1. Initial program 74.3%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 81.5

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

   4. Applied rewrites 81.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 81.4

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

   6. Applied rewrites 81.4%

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

Alternative 4: 76.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(b \cdot {\left(\frac{d}{c} \cdot -1\right)}^{2}, -1, \frac{d \cdot a}{c} \cdot -1\right) + b}{c}\\ \mathbf{if}\;c \leq -350000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 12.6:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0
         (/
          (+
           (fma (* b (pow (* (/ d c) -1.0) 2.0)) -1.0 (* (/ (* d a) c) -1.0))
           b)
          c)))
   (if (<= c -350000000.0)
     t_0
     (if (<= c 12.6) (/ (fma b (/ c d) (* -1.0 a)) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (fma((b * pow(((d / c) * -1.0), 2.0)), -1.0, (((d * a) / c) * -1.0)) + b) / c;
	double tmp;
	if (c <= -350000000.0) {
		tmp = t_0;
	} else if (c <= 12.6) {
		tmp = fma(b, (c / d), (-1.0 * a)) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(fma(Float64(b * (Float64(Float64(d / c) * -1.0) ^ 2.0)), -1.0, Float64(Float64(Float64(d * a) / c) * -1.0)) + b) / c)
	tmp = 0.0
	if (c <= -350000000.0)
		tmp = t_0;
	elseif (c <= 12.6)
		tmp = Float64(fma(b, Float64(c / d), Float64(-1.0 * a)) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(N[(b * N[Power[N[(N[(d / c), $MachinePrecision] * -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -350000000.0], t$95$0, If[LessEqual[c, 12.6], N[(N[(b * N[(c / d), $MachinePrecision] + N[(-1.0 * a), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -3.5e8 or 12.5999999999999996 < c

   1. Initial program 51.7%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 24.0

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

   4. Applied rewrites 24.0%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 24.8

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

   7. Applied rewrites 24.8%

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

   8. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 71.6%

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

   if -3.5e8 < c < 12.5999999999999996

   1. Initial program 74.6%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 80.2

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

   4. Applied rewrites 80.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 80.1

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

   6. Applied rewrites 80.1%

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

Alternative 5: 76.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(b \cdot {\left(\frac{d}{c} \cdot -1\right)}^{2}, -1, \frac{d \cdot a}{c} \cdot -1\right) + b}{c}\\ \mathbf{if}\;c \leq -350000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 12.6:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, a, \frac{c \cdot b}{d}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0
         (/
          (+
           (fma (* b (pow (* (/ d c) -1.0) 2.0)) -1.0 (* (/ (* d a) c) -1.0))
           b)
          c)))
   (if (<= c -350000000.0)
     t_0
     (if (<= c 12.6) (/ (fma -1.0 a (/ (* c b) d)) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (fma((b * pow(((d / c) * -1.0), 2.0)), -1.0, (((d * a) / c) * -1.0)) + b) / c;
	double tmp;
	if (c <= -350000000.0) {
		tmp = t_0;
	} else if (c <= 12.6) {
		tmp = fma(-1.0, a, ((c * b) / d)) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(fma(Float64(b * (Float64(Float64(d / c) * -1.0) ^ 2.0)), -1.0, Float64(Float64(Float64(d * a) / c) * -1.0)) + b) / c)
	tmp = 0.0
	if (c <= -350000000.0)
		tmp = t_0;
	elseif (c <= 12.6)
		tmp = Float64(fma(-1.0, a, Float64(Float64(c * b) / d)) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(N[(b * N[Power[N[(N[(d / c), $MachinePrecision] * -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -350000000.0], t$95$0, If[LessEqual[c, 12.6], N[(N[(-1.0 * a + N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -3.5e8 or 12.5999999999999996 < c

   1. Initial program 51.7%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 24.0

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

   4. Applied rewrites 24.0%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 24.8

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

   7. Applied rewrites 24.8%

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

   8. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 71.6%

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

   if -3.5e8 < c < 12.5999999999999996

   1. Initial program 74.6%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 80.2

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

   4. Applied rewrites 80.2%

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

Alternative 6: 76.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{c}{d}, b, a \cdot {\left(\frac{c}{d} \cdot -1\right)}^{2}\right) + -1 \cdot a}{d}\\ \mathbf{if}\;d \leq -6.5 \cdot 10^{+49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot {\left(\frac{d}{c} \cdot -1\right)}^{2}, -1, \frac{d \cdot a}{c} \cdot -1\right) + b}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0
         (/
          (+ (fma (/ c d) b (* a (pow (* (/ c d) -1.0) 2.0))) (* -1.0 a))
          d)))
   (if (<= d -6.5e+49)
     t_0
     (if (<= d 2.3e+62)
       (/
        (+
         (fma (* b (pow (* (/ d c) -1.0) 2.0)) -1.0 (* (/ (* d a) c) -1.0))
         b)
        c)
       t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (fma((c / d), b, (a * pow(((c / d) * -1.0), 2.0))) + (-1.0 * a)) / d;
	double tmp;
	if (d <= -6.5e+49) {
		tmp = t_0;
	} else if (d <= 2.3e+62) {
		tmp = (fma((b * pow(((d / c) * -1.0), 2.0)), -1.0, (((d * a) / c) * -1.0)) + b) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(fma(Float64(c / d), b, Float64(a * (Float64(Float64(c / d) * -1.0) ^ 2.0))) + Float64(-1.0 * a)) / d)
	tmp = 0.0
	if (d <= -6.5e+49)
		tmp = t_0;
	elseif (d <= 2.3e+62)
		tmp = Float64(Float64(fma(Float64(b * (Float64(Float64(d / c) * -1.0) ^ 2.0)), -1.0, Float64(Float64(Float64(d * a) / c) * -1.0)) + b) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(N[(c / d), $MachinePrecision] * b + N[(a * N[Power[N[(N[(c / d), $MachinePrecision] * -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * a), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -6.5e+49], t$95$0, If[LessEqual[d, 2.3e+62], N[(N[(N[(N[(b * N[Power[N[(N[(d / c), $MachinePrecision] * -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -6.5000000000000005e49 or 2.29999999999999984e62 < d

   1. Initial program 46.5%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 77.1

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 72.2

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

   7. Applied rewrites 72.2%

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

   8. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 80.3%

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

   if -6.5000000000000005e49 < d < 2.29999999999999984e62

   1. Initial program 75.3%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 35.7

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

   4. Applied rewrites 35.7%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 33.0

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

   7. Applied rewrites 33.0%

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

   8. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 72.9%

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

Alternative 7: 74.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.02 \cdot 10^{+48}:\\ \;\;\;\;\frac{\left(b \cdot \frac{c}{d \cdot a} - 1\right) \cdot a}{d}\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot {\left(\frac{d}{c} \cdot -1\right)}^{2}, -1, \frac{d \cdot a}{c} \cdot -1\right) + b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{b}{a} \cdot \frac{c}{d} - 1\right) \cdot a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.02e+48)
   (/ (* (- (* b (/ c (* d a))) 1.0) a) d)
   (if (<= d 2.3e+62)
     (/
      (+ (fma (* b (pow (* (/ d c) -1.0) 2.0)) -1.0 (* (/ (* d a) c) -1.0)) b)
      c)
     (/ (* (- (* (/ b a) (/ c d)) 1.0) a) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.02e+48) {
		tmp = (((b * (c / (d * a))) - 1.0) * a) / d;
	} else if (d <= 2.3e+62) {
		tmp = (fma((b * pow(((d / c) * -1.0), 2.0)), -1.0, (((d * a) / c) * -1.0)) + b) / c;
	} else {
		tmp = ((((b / a) * (c / d)) - 1.0) * a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.02e+48)
		tmp = Float64(Float64(Float64(Float64(b * Float64(c / Float64(d * a))) - 1.0) * a) / d);
	elseif (d <= 2.3e+62)
		tmp = Float64(Float64(fma(Float64(b * (Float64(Float64(d / c) * -1.0) ^ 2.0)), -1.0, Float64(Float64(Float64(d * a) / c) * -1.0)) + b) / c);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(b / a) * Float64(c / d)) - 1.0) * a) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.02e+48], N[(N[(N[(N[(b * N[(c / N[(d * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 2.3e+62], N[(N[(N[(N[(b * N[Power[N[(N[(d / c), $MachinePrecision] * -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(N[(N[(b / a), $MachinePrecision] * N[(c / d), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * a), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.02e48

   1. Initial program 47.9%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 77.0

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

   4. Applied rewrites 77.0%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 71.7

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

   7. Applied rewrites 71.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 78.6

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

   9. Applied rewrites 78.6%

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

   if -1.02e48 < d < 2.29999999999999984e62

   1. Initial program 75.3%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 35.6

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

   4. Applied rewrites 35.6%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 33.0

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

   7. Applied rewrites 33.0%

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

   8. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 73.0%

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

   if 2.29999999999999984e62 < d

   1. Initial program 45.3%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 77.0

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

   4. Applied rewrites 77.0%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 72.5

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

   7. Applied rewrites 72.5%

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

Alternative 8: 51.8% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.4%

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

2. Taylor expanded in d around inf

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 52.7

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

4. Applied rewrites 52.7%

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

5. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. times-frac N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 49.2

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

7. Applied rewrites 49.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 51.8

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

9. Applied rewrites 51.8%

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

Reproduce

herbie shell --seed 2025093 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64
  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))