Rosa's DopplerBench

Percentage Accurate: 72.6% → 99.1%

Time: 4.3s

Alternatives: 10

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))

C

double code(double u, double v, double t1) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}

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(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (-t1 * v) / ((t1 + u) * (t1 + u))
end function

Java

public static double code(double u, double v, double t1) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}

Python

def code(u, v, t1):
	return (-t1 * v) / ((t1 + u) * (t1 + u))

Julia

function code(u, v, t1)
	return Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
end

Wolfram

code[u_, v_, t1_] := N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))

C

double code(double u, double v, double t1) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}

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(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (-t1 * v) / ((t1 + u) * (t1 + u))
end function

Java

public static double code(double u, double v, double t1) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}

Python

def code(u, v, t1):
	return (-t1 * v) / ((t1 + u) * (t1 + u))

Julia

function code(u, v, t1)
	return Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
end

Wolfram

code[u_, v_, t1_] := N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ \begin{array}{l} t_1 := \left(-t1\right) - u\\ v\_s \cdot \begin{array}{l} \mathbf{if}\;v\_m \leq 3.4 \cdot 10^{+125}:\\ \;\;\;\;\frac{v\_m}{t\_1} \cdot \frac{t1}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{t\_1} \cdot v\_m}{u + t1}\\ \end{array} \end{array} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1)
 :precision binary64
 (let* ((t_1 (- (- t1) u)))
   (*
    v_s
    (if (<= v_m 3.4e+125)
      (* (/ v_m t_1) (/ t1 (+ u t1)))
      (/ (* (/ t1 t_1) v_m) (+ u t1))))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	double t_1 = -t1 - u;
	double tmp;
	if (v_m <= 3.4e+125) {
		tmp = (v_m / t_1) * (t1 / (u + t1));
	} else {
		tmp = ((t1 / t_1) * v_m) / (u + t1);
	}
	return v_s * tmp;
}

Fortran

v\_m =     private
v\_s =     private
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(v_s, u, v_m, t1)
use fmin_fmax_functions
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -t1 - u
    if (v_m <= 3.4d+125) then
        tmp = (v_m / t_1) * (t1 / (u + t1))
    else
        tmp = ((t1 / t_1) * v_m) / (u + t1)
    end if
    code = v_s * tmp
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	double t_1 = -t1 - u;
	double tmp;
	if (v_m <= 3.4e+125) {
		tmp = (v_m / t_1) * (t1 / (u + t1));
	} else {
		tmp = ((t1 / t_1) * v_m) / (u + t1);
	}
	return v_s * tmp;
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	t_1 = -t1 - u
	tmp = 0
	if v_m <= 3.4e+125:
		tmp = (v_m / t_1) * (t1 / (u + t1))
	else:
		tmp = ((t1 / t_1) * v_m) / (u + t1)
	return v_s * tmp

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	t_1 = Float64(Float64(-t1) - u)
	tmp = 0.0
	if (v_m <= 3.4e+125)
		tmp = Float64(Float64(v_m / t_1) * Float64(t1 / Float64(u + t1)));
	else
		tmp = Float64(Float64(Float64(t1 / t_1) * v_m) / Float64(u + t1));
	end
	return Float64(v_s * tmp)
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp_2 = code(v_s, u, v_m, t1)
	t_1 = -t1 - u;
	tmp = 0.0;
	if (v_m <= 3.4e+125)
		tmp = (v_m / t_1) * (t1 / (u + t1));
	else
		tmp = ((t1 / t_1) * v_m) / (u + t1);
	end
	tmp_2 = v_s * tmp;
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := Block[{t$95$1 = N[((-t1) - u), $MachinePrecision]}, N[(v$95$s * If[LessEqual[v$95$m, 3.4e+125], N[(N[(v$95$m / t$95$1), $MachinePrecision] * N[(t1 / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t1 / t$95$1), $MachinePrecision] * v$95$m), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 3.3999999999999999e125

   1. Initial program 72.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-t1\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{v \cdot \left(-t1\right)}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      5. sqr-neg-rev N/A

         \[\leadsto \frac{v \cdot \left(-t1\right)}{\color{blue}{\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{v}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{v}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{v}{\mathsf{neg}\left(\left(t1 + u\right)\right)}} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{v}{\mathsf{neg}\left(\color{blue}{\left(t1 + u\right)}\right)} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

      10. distribute-neg-in N/A

          \[\leadsto \frac{v}{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) + \left(\mathsf{neg}\left(u\right)\right)}} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{v}{\color{blue}{\left(-t1\right)} + \left(\mathsf{neg}\left(u\right)\right)} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

      12. sub-flip-reverse N/A

          \[\leadsto \frac{v}{\color{blue}{\left(-t1\right) - u}} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{v}{\color{blue}{\left(-t1\right) - u}} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

      14. lift-neg.f64 N/A

          \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \frac{\color{blue}{\mathsf{neg}\left(t1\right)}}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

      15. frac-2neg-rev N/A

          \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \color{blue}{\frac{t1}{t1 + u}} \]

      16. lower-/.f64 98.2

          \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \color{blue}{\frac{t1}{t1 + u}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \frac{t1}{\color{blue}{t1 + u}} \]

      18. +-commutative N/A

          \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \frac{t1}{\color{blue}{u + t1}} \]

      19. lower-+.f64 98.2

          \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \frac{t1}{\color{blue}{u + t1}} \]

   3. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\frac{v}{\left(-t1\right) - u} \cdot \frac{t1}{u + t1}} \]

   if 3.3999999999999999e125 < v

   1. Initial program 72.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(-t1\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(-t1\right) \cdot v}}{t1 + u}}{t1 + u} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{-t1}{t1 + u} \cdot v}}{t1 + u} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{-t1}{t1 + u} \cdot v}}{t1 + u} \]

      8. frac-2neg N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(\left(-t1\right)\right)}{\mathsf{neg}\left(\left(t1 + u\right)\right)}} \cdot v}{t1 + u} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)}\right)}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \cdot v}{t1 + u} \]

      10. remove-double-neg N/A

          \[\leadsto \frac{\frac{\color{blue}{t1}}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \cdot v}{t1 + u} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)}} \cdot v}{t1 + u} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{t1}{\mathsf{neg}\left(\color{blue}{\left(t1 + u\right)}\right)} \cdot v}{t1 + u} \]

      13. distribute-neg-in N/A

          \[\leadsto \frac{\frac{t1}{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) + \left(\mathsf{neg}\left(u\right)\right)}} \cdot v}{t1 + u} \]

      14. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{t1}{\color{blue}{\left(-t1\right)} + \left(\mathsf{neg}\left(u\right)\right)} \cdot v}{t1 + u} \]

      15. sub-flip-reverse N/A

          \[\leadsto \frac{\frac{t1}{\color{blue}{\left(-t1\right) - u}} \cdot v}{t1 + u} \]

      16. lower--.f64 98.3

          \[\leadsto \frac{\frac{t1}{\color{blue}{\left(-t1\right) - u}} \cdot v}{t1 + u} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\frac{t1}{\left(-t1\right) - u} \cdot v}{\color{blue}{t1 + u}} \]

      18. +-commutative N/A

          \[\leadsto \frac{\frac{t1}{\left(-t1\right) - u} \cdot v}{\color{blue}{u + t1}} \]

      19. lower-+.f64 98.3

          \[\leadsto \frac{\frac{t1}{\left(-t1\right) - u} \cdot v}{\color{blue}{u + t1}} \]

   3. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\frac{\frac{t1}{\left(-t1\right) - u} \cdot v}{u + t1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \left(\frac{v\_m}{\left(-t1\right) - u} \cdot \frac{t1}{u + t1}\right) \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1)
 :precision binary64
 (* v_s (* (/ v_m (- (- t1) u)) (/ t1 (+ u t1)))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	return v_s * ((v_m / (-t1 - u)) * (t1 / (u + t1)));
}

Fortran

v\_m =     private
v\_s =     private
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(v_s, u, v_m, t1)
use fmin_fmax_functions
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    code = v_s * ((v_m / (-t1 - u)) * (t1 / (u + t1)))
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	return v_s * ((v_m / (-t1 - u)) * (t1 / (u + t1)));
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	return v_s * ((v_m / (-t1 - u)) * (t1 / (u + t1)))

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	return Float64(v_s * Float64(Float64(v_m / Float64(Float64(-t1) - u)) * Float64(t1 / Float64(u + t1))))
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp = code(v_s, u, v_m, t1)
	tmp = v_s * ((v_m / (-t1 - u)) * (t1 / (u + t1)));
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * N[(N[(v$95$m / N[((-t1) - u), $MachinePrecision]), $MachinePrecision] * N[(t1 / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.6%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-t1\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{v \cdot \left(-t1\right)}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   5. sqr-neg-rev N/A

      \[\leadsto \frac{v \cdot \left(-t1\right)}{\color{blue}{\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right)}} \]

   6. times-frac N/A

      \[\leadsto \color{blue}{\frac{v}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{v}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{v}{\mathsf{neg}\left(\left(t1 + u\right)\right)}} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

   9. lift-+.f64 N/A

      \[\leadsto \frac{v}{\mathsf{neg}\left(\color{blue}{\left(t1 + u\right)}\right)} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

   10. distribute-neg-in N/A

       \[\leadsto \frac{v}{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) + \left(\mathsf{neg}\left(u\right)\right)}} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

   11. lift-neg.f64 N/A

       \[\leadsto \frac{v}{\color{blue}{\left(-t1\right)} + \left(\mathsf{neg}\left(u\right)\right)} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

   12. sub-flip-reverse N/A

       \[\leadsto \frac{v}{\color{blue}{\left(-t1\right) - u}} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

   13. lower--.f64 N/A

       \[\leadsto \frac{v}{\color{blue}{\left(-t1\right) - u}} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

   14. lift-neg.f64 N/A

       \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \frac{\color{blue}{\mathsf{neg}\left(t1\right)}}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

   15. frac-2neg-rev N/A

       \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \color{blue}{\frac{t1}{t1 + u}} \]

   16. lower-/.f64 98.2

       \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \color{blue}{\frac{t1}{t1 + u}} \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \frac{t1}{\color{blue}{t1 + u}} \]

   18. +-commutative N/A

       \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \frac{t1}{\color{blue}{u + t1}} \]

   19. lower-+.f64 98.2

       \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \frac{t1}{\color{blue}{u + t1}} \]

3. Applied rewrites 98.2%

   \[\leadsto \color{blue}{\frac{v}{\left(-t1\right) - u} \cdot \frac{t1}{u + t1}} \]
4. Add Preprocessing

Alternative 3: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \frac{\frac{v\_m}{u + t1}}{-1 - \frac{u}{t1}} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1)
 :precision binary64
 (* v_s (/ (/ v_m (+ u t1)) (- -1.0 (/ u t1)))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	return v_s * ((v_m / (u + t1)) / (-1.0 - (u / t1)));
}

Fortran

v\_m =     private
v\_s =     private
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(v_s, u, v_m, t1)
use fmin_fmax_functions
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    code = v_s * ((v_m / (u + t1)) / ((-1.0d0) - (u / t1)))
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	return v_s * ((v_m / (u + t1)) / (-1.0 - (u / t1)));
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	return v_s * ((v_m / (u + t1)) / (-1.0 - (u / t1)))

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	return Float64(v_s * Float64(Float64(v_m / Float64(u + t1)) / Float64(-1.0 - Float64(u / t1))))
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp = code(v_s, u, v_m, t1)
	tmp = v_s * ((v_m / (u + t1)) / (-1.0 - (u / t1)));
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * N[(N[(v$95$m / N[(u + t1), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.6%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   2. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{\left(-t1\right) \cdot v}}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{\color{blue}{\left(-t1\right) \cdot v}}} \]

   4. associate-/r* N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}{v}}} \]

   5. div-flip-rev N/A

      \[\leadsto \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{v}{\frac{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}}{-t1}} \]

   8. sqr-neg-rev N/A

      \[\leadsto \frac{v}{\frac{\color{blue}{\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right)}}{-t1}} \]

   9. associate-/l* N/A

      \[\leadsto \frac{v}{\color{blue}{\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right) \cdot \frac{\mathsf{neg}\left(\left(t1 + u\right)\right)}{-t1}}} \]

   10. frac-2neg N/A

       \[\leadsto \frac{v}{\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right)\right)}{\mathsf{neg}\left(\left(-t1\right)\right)}}} \]

3. Applied rewrites 95.3%

   \[\leadsto \color{blue}{\frac{v}{\left(\left(-t1\right) - u\right) \cdot \left(\frac{u}{t1} - -1\right)}} \]
4. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{v}{\left(\left(-t1\right) - u\right) \cdot \color{blue}{\left(\frac{u}{t1} - -1\right)}} \]

   2. sub-flip N/A

      \[\leadsto \frac{v}{\left(\left(-t1\right) - u\right) \cdot \color{blue}{\left(\frac{u}{t1} + \left(\mathsf{neg}\left(-1\right)\right)\right)}} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{v}{\left(\left(-t1\right) - u\right) \cdot \left(\color{blue}{\frac{u}{t1}} + \left(\mathsf{neg}\left(-1\right)\right)\right)} \]

   4. mult-flip N/A

      \[\leadsto \frac{v}{\left(\left(-t1\right) - u\right) \cdot \left(\color{blue}{u \cdot \frac{1}{t1}} + \left(\mathsf{neg}\left(-1\right)\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{v}{\left(\left(-t1\right) - u\right) \cdot \left(\color{blue}{\frac{1}{t1} \cdot u} + \left(\mathsf{neg}\left(-1\right)\right)\right)} \]

   6. mul-1-neg N/A

      \[\leadsto \frac{v}{\left(\left(-t1\right) - u\right) \cdot \left(\frac{1}{t1} \cdot u + \color{blue}{-1 \cdot -1}\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{v}{\left(\left(-t1\right) - u\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{t1}, u, -1 \cdot -1\right)}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{v}{\left(\left(-t1\right) - u\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{-1 \cdot -1}}{t1}, u, -1 \cdot -1\right)} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{v}{\left(\left(-t1\right) - u\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot -1}{t1}}, u, -1 \cdot -1\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{v}{\left(\left(-t1\right) - u\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{1}}{t1}, u, -1 \cdot -1\right)} \]

   11. metadata-eval 95.3

       \[\leadsto \frac{v}{\left(\left(-t1\right) - u\right) \cdot \mathsf{fma}\left(\frac{1}{t1}, u, \color{blue}{1}\right)} \]

5. Applied rewrites 95.3%

   \[\leadsto \frac{v}{\left(\left(-t1\right) - u\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{t1}, u, 1\right)}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{v}{\left(\left(-t1\right) - u\right) \cdot \mathsf{fma}\left(\frac{1}{t1}, u, 1\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{v}{\color{blue}{\left(\left(-t1\right) - u\right) \cdot \mathsf{fma}\left(\frac{1}{t1}, u, 1\right)}} \]

   3. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{v}{\left(-t1\right) - u}}{\mathsf{fma}\left(\frac{1}{t1}, u, 1\right)}} \]

   4. frac-2neg-rev N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(v\right)}{\mathsf{neg}\left(\left(\left(-t1\right) - u\right)\right)}}}{\mathsf{fma}\left(\frac{1}{t1}, u, 1\right)} \]

   5. lift--.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{neg}\left(v\right)}{\mathsf{neg}\left(\color{blue}{\left(\left(-t1\right) - u\right)}\right)}}{\mathsf{fma}\left(\frac{1}{t1}, u, 1\right)} \]

   6. sub-negate-rev N/A

      \[\leadsto \frac{\frac{\mathsf{neg}\left(v\right)}{\color{blue}{u - \left(-t1\right)}}}{\mathsf{fma}\left(\frac{1}{t1}, u, 1\right)} \]

   7. lift-neg.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{neg}\left(v\right)}{u - \color{blue}{\left(\mathsf{neg}\left(t1\right)\right)}}}{\mathsf{fma}\left(\frac{1}{t1}, u, 1\right)} \]

   8. add-flip N/A

      \[\leadsto \frac{\frac{\mathsf{neg}\left(v\right)}{\color{blue}{u + t1}}}{\mathsf{fma}\left(\frac{1}{t1}, u, 1\right)} \]

   9. +-commutative N/A

      \[\leadsto \frac{\frac{\mathsf{neg}\left(v\right)}{\color{blue}{t1 + u}}}{\mathsf{fma}\left(\frac{1}{t1}, u, 1\right)} \]

   10. distribute-frac-neg N/A

       \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{v}{t1 + u}\right)}}{\mathsf{fma}\left(\frac{1}{t1}, u, 1\right)} \]

   11. lift-fma.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(\frac{v}{t1 + u}\right)}{\color{blue}{\frac{1}{t1} \cdot u + 1}} \]

   12. add-flip N/A

       \[\leadsto \frac{\mathsf{neg}\left(\frac{v}{t1 + u}\right)}{\color{blue}{\frac{1}{t1} \cdot u - \left(\mathsf{neg}\left(1\right)\right)}} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\mathsf{neg}\left(\frac{v}{t1 + u}\right)}{\frac{1}{t1} \cdot u - \color{blue}{-1}} \]

   14. sub-negate-rev N/A

       \[\leadsto \frac{\mathsf{neg}\left(\frac{v}{t1 + u}\right)}{\color{blue}{\mathsf{neg}\left(\left(-1 - \frac{1}{t1} \cdot u\right)\right)}} \]

   15. frac-2neg-rev N/A

       \[\leadsto \color{blue}{\frac{\frac{v}{t1 + u}}{-1 - \frac{1}{t1} \cdot u}} \]

   16. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{v}{t1 + u}}{-1 - \frac{1}{t1} \cdot u}} \]

   17. lower-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{v}{t1 + u}}}{-1 - \frac{1}{t1} \cdot u} \]

   18. +-commutative N/A

       \[\leadsto \frac{\frac{v}{\color{blue}{u + t1}}}{-1 - \frac{1}{t1} \cdot u} \]

   19. lift-+.f64 N/A

       \[\leadsto \frac{\frac{v}{\color{blue}{u + t1}}}{-1 - \frac{1}{t1} \cdot u} \]

   20. lower--.f64 N/A

       \[\leadsto \frac{\frac{v}{u + t1}}{\color{blue}{-1 - \frac{1}{t1} \cdot u}} \]

   21. *-commutative N/A

       \[\leadsto \frac{\frac{v}{u + t1}}{-1 - \color{blue}{u \cdot \frac{1}{t1}}} \]

   22. lift-/.f64 N/A

       \[\leadsto \frac{\frac{v}{u + t1}}{-1 - u \cdot \color{blue}{\frac{1}{t1}}} \]

   23. mult-flip-rev N/A

       \[\leadsto \frac{\frac{v}{u + t1}}{-1 - \color{blue}{\frac{u}{t1}}} \]

   24. lower-/.f64 98.1

       \[\leadsto \frac{\frac{v}{u + t1}}{-1 - \color{blue}{\frac{u}{t1}}} \]

7. Applied rewrites 98.1%

   \[\leadsto \color{blue}{\frac{\frac{v}{u + t1}}{-1 - \frac{u}{t1}}} \]
8. Add Preprocessing

Alternative 4: 79.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \begin{array}{l} \mathbf{if}\;t1 \leq -2.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{v\_m}{t1 \cdot \left(-2 \cdot \frac{u}{t1} - 1\right)}\\ \mathbf{elif}\;t1 \leq 1500000000000:\\ \;\;\;\;\left(-1 \cdot \frac{v\_m}{u}\right) \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v\_m}{\mathsf{fma}\left(-2, u, -1 \cdot t1\right)}\\ \end{array} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1)
 :precision binary64
 (*
  v_s
  (if (<= t1 -2.2e-45)
    (/ v_m (* t1 (- (* -2.0 (/ u t1)) 1.0)))
    (if (<= t1 1500000000000.0)
      (* (* -1.0 (/ v_m u)) (/ t1 u))
      (/ v_m (fma -2.0 u (* -1.0 t1)))))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	double tmp;
	if (t1 <= -2.2e-45) {
		tmp = v_m / (t1 * ((-2.0 * (u / t1)) - 1.0));
	} else if (t1 <= 1500000000000.0) {
		tmp = (-1.0 * (v_m / u)) * (t1 / u);
	} else {
		tmp = v_m / fma(-2.0, u, (-1.0 * t1));
	}
	return v_s * tmp;
}

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	tmp = 0.0
	if (t1 <= -2.2e-45)
		tmp = Float64(v_m / Float64(t1 * Float64(Float64(-2.0 * Float64(u / t1)) - 1.0)));
	elseif (t1 <= 1500000000000.0)
		tmp = Float64(Float64(-1.0 * Float64(v_m / u)) * Float64(t1 / u));
	else
		tmp = Float64(v_m / fma(-2.0, u, Float64(-1.0 * t1)));
	end
	return Float64(v_s * tmp)
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * If[LessEqual[t1, -2.2e-45], N[(v$95$m / N[(t1 * N[(N[(-2.0 * N[(u / t1), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1500000000000.0], N[(N[(-1.0 * N[(v$95$m / u), $MachinePrecision]), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], N[(v$95$m / N[(-2.0 * u + N[(-1.0 * t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.19999999999999993e-45

   1. Initial program 72.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{\left(-t1\right) \cdot v}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{\color{blue}{\left(-t1\right) \cdot v}}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}{v}}} \]

      5. div-flip-rev N/A

         \[\leadsto \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{v}{\frac{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}}{-t1}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \frac{v}{\frac{\color{blue}{\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right)}}{-t1}} \]

      9. associate-/l* N/A

         \[\leadsto \frac{v}{\color{blue}{\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right) \cdot \frac{\mathsf{neg}\left(\left(t1 + u\right)\right)}{-t1}}} \]

      10. frac-2neg N/A

          \[\leadsto \frac{v}{\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right)\right)}{\mathsf{neg}\left(\left(-t1\right)\right)}}} \]

   3. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\frac{v}{\left(\left(-t1\right) - u\right) \cdot \left(\frac{u}{t1} - -1\right)}} \]

   4. Taylor expanded in t1 around inf

      \[\leadsto \frac{v}{\color{blue}{t1 \cdot \left(-2 \cdot \frac{u}{t1} - 1\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{v}{t1 \cdot \color{blue}{\left(-2 \cdot \frac{u}{t1} - 1\right)}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{v}{t1 \cdot \left(-2 \cdot \frac{u}{t1} - \color{blue}{1}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{v}{t1 \cdot \left(-2 \cdot \frac{u}{t1} - 1\right)} \]

      4. lower-/.f64 69.3

         \[\leadsto \frac{v}{t1 \cdot \left(-2 \cdot \frac{u}{t1} - 1\right)} \]

   6. Applied rewrites 69.3%

      \[\leadsto \frac{v}{\color{blue}{t1 \cdot \left(-2 \cdot \frac{u}{t1} - 1\right)}} \]

   if -2.19999999999999993e-45 < t1 < 1.5e12

   1. Initial program 72.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-t1\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{v \cdot \left(-t1\right)}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      5. sqr-neg-rev N/A

         \[\leadsto \frac{v \cdot \left(-t1\right)}{\color{blue}{\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{v}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{v}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{v}{\mathsf{neg}\left(\left(t1 + u\right)\right)}} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{v}{\mathsf{neg}\left(\color{blue}{\left(t1 + u\right)}\right)} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

      10. distribute-neg-in N/A

          \[\leadsto \frac{v}{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) + \left(\mathsf{neg}\left(u\right)\right)}} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{v}{\color{blue}{\left(-t1\right)} + \left(\mathsf{neg}\left(u\right)\right)} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

      12. sub-flip-reverse N/A

          \[\leadsto \frac{v}{\color{blue}{\left(-t1\right) - u}} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{v}{\color{blue}{\left(-t1\right) - u}} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

      14. lift-neg.f64 N/A

          \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \frac{\color{blue}{\mathsf{neg}\left(t1\right)}}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

      15. frac-2neg-rev N/A

          \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \color{blue}{\frac{t1}{t1 + u}} \]

      16. lower-/.f64 98.2

          \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \color{blue}{\frac{t1}{t1 + u}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \frac{t1}{\color{blue}{t1 + u}} \]

      18. +-commutative N/A

          \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \frac{t1}{\color{blue}{u + t1}} \]

      19. lower-+.f64 98.2

          \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \frac{t1}{\color{blue}{u + t1}} \]

   3. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\frac{v}{\left(-t1\right) - u} \cdot \frac{t1}{u + t1}} \]

   4. Taylor expanded in u around inf

      \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \color{blue}{\frac{t1}{u}} \]
   5. Step-by-step derivation

      1. lower-/.f64 54.6

         \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \frac{t1}{\color{blue}{u}} \]

   6. Applied rewrites 54.6%

      \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \color{blue}{\frac{t1}{u}} \]

   7. Taylor expanded in u around inf

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{v}{u}\right)} \cdot \frac{t1}{u} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\frac{v}{u}}\right) \cdot \frac{t1}{u} \]

      2. lower-/.f64 52.3

         \[\leadsto \left(-1 \cdot \frac{v}{\color{blue}{u}}\right) \cdot \frac{t1}{u} \]

   9. Applied rewrites 52.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{v}{u}\right)} \cdot \frac{t1}{u} \]

   if 1.5e12 < t1

   1. Initial program 72.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{\left(-t1\right) \cdot v}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{\color{blue}{\left(-t1\right) \cdot v}}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}{v}}} \]

      5. div-flip-rev N/A

         \[\leadsto \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{v}{\frac{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}}{-t1}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \frac{v}{\frac{\color{blue}{\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right)}}{-t1}} \]

      9. associate-/l* N/A

         \[\leadsto \frac{v}{\color{blue}{\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right) \cdot \frac{\mathsf{neg}\left(\left(t1 + u\right)\right)}{-t1}}} \]

      10. frac-2neg N/A

          \[\leadsto \frac{v}{\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right)\right)}{\mathsf{neg}\left(\left(-t1\right)\right)}}} \]

   3. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\frac{v}{\left(\left(-t1\right) - u\right) \cdot \left(\frac{u}{t1} - -1\right)}} \]

   4. Taylor expanded in u around 0

      \[\leadsto \frac{v}{\color{blue}{-2 \cdot u + -1 \cdot t1}} \]
   5. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \frac{v}{\mathsf{fma}\left(-2, \color{blue}{u}, -1 \cdot t1\right)} \]

      2. lower-*.f64 62.7

         \[\leadsto \frac{v}{\mathsf{fma}\left(-2, u, -1 \cdot t1\right)} \]

   6. Applied rewrites 62.7%

      \[\leadsto \frac{v}{\color{blue}{\mathsf{fma}\left(-2, u, -1 \cdot t1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 79.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ \begin{array}{l} t_1 := \frac{v\_m}{\mathsf{fma}\left(-2, u, -1 \cdot t1\right)}\\ v\_s \cdot \begin{array}{l} \mathbf{if}\;t1 \leq -3.3 \cdot 10^{-45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1500000000000:\\ \;\;\;\;\left(-1 \cdot \frac{v\_m}{u}\right) \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1)
 :precision binary64
 (let* ((t_1 (/ v_m (fma -2.0 u (* -1.0 t1)))))
   (*
    v_s
    (if (<= t1 -3.3e-45)
      t_1
      (if (<= t1 1500000000000.0) (* (* -1.0 (/ v_m u)) (/ t1 u)) t_1)))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	double t_1 = v_m / fma(-2.0, u, (-1.0 * t1));
	double tmp;
	if (t1 <= -3.3e-45) {
		tmp = t_1;
	} else if (t1 <= 1500000000000.0) {
		tmp = (-1.0 * (v_m / u)) * (t1 / u);
	} else {
		tmp = t_1;
	}
	return v_s * tmp;
}

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	t_1 = Float64(v_m / fma(-2.0, u, Float64(-1.0 * t1)))
	tmp = 0.0
	if (t1 <= -3.3e-45)
		tmp = t_1;
	elseif (t1 <= 1500000000000.0)
		tmp = Float64(Float64(-1.0 * Float64(v_m / u)) * Float64(t1 / u));
	else
		tmp = t_1;
	end
	return Float64(v_s * tmp)
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := Block[{t$95$1 = N[(v$95$m / N[(-2.0 * u + N[(-1.0 * t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(v$95$s * If[LessEqual[t1, -3.3e-45], t$95$1, If[LessEqual[t1, 1500000000000.0], N[(N[(-1.0 * N[(v$95$m / u), $MachinePrecision]), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.3000000000000001e-45 or 1.5e12 < t1

   1. Initial program 72.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{\left(-t1\right) \cdot v}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{\color{blue}{\left(-t1\right) \cdot v}}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}{v}}} \]

      5. div-flip-rev N/A

         \[\leadsto \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{v}{\frac{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}}{-t1}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \frac{v}{\frac{\color{blue}{\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right)}}{-t1}} \]

      9. associate-/l* N/A

         \[\leadsto \frac{v}{\color{blue}{\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right) \cdot \frac{\mathsf{neg}\left(\left(t1 + u\right)\right)}{-t1}}} \]

      10. frac-2neg N/A

          \[\leadsto \frac{v}{\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right)\right)}{\mathsf{neg}\left(\left(-t1\right)\right)}}} \]

   3. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\frac{v}{\left(\left(-t1\right) - u\right) \cdot \left(\frac{u}{t1} - -1\right)}} \]

   4. Taylor expanded in u around 0

      \[\leadsto \frac{v}{\color{blue}{-2 \cdot u + -1 \cdot t1}} \]
   5. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \frac{v}{\mathsf{fma}\left(-2, \color{blue}{u}, -1 \cdot t1\right)} \]

      2. lower-*.f64 62.7

         \[\leadsto \frac{v}{\mathsf{fma}\left(-2, u, -1 \cdot t1\right)} \]

   6. Applied rewrites 62.7%

      \[\leadsto \frac{v}{\color{blue}{\mathsf{fma}\left(-2, u, -1 \cdot t1\right)}} \]

   if -3.3000000000000001e-45 < t1 < 1.5e12

   1. Initial program 72.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-t1\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{v \cdot \left(-t1\right)}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      5. sqr-neg-rev N/A

         \[\leadsto \frac{v \cdot \left(-t1\right)}{\color{blue}{\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{v}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{v}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{v}{\mathsf{neg}\left(\left(t1 + u\right)\right)}} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{v}{\mathsf{neg}\left(\color{blue}{\left(t1 + u\right)}\right)} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

      10. distribute-neg-in N/A

          \[\leadsto \frac{v}{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) + \left(\mathsf{neg}\left(u\right)\right)}} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{v}{\color{blue}{\left(-t1\right)} + \left(\mathsf{neg}\left(u\right)\right)} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

      12. sub-flip-reverse N/A

          \[\leadsto \frac{v}{\color{blue}{\left(-t1\right) - u}} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{v}{\color{blue}{\left(-t1\right) - u}} \cdot \frac{-t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

      14. lift-neg.f64 N/A

          \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \frac{\color{blue}{\mathsf{neg}\left(t1\right)}}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \]

      15. frac-2neg-rev N/A

          \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \color{blue}{\frac{t1}{t1 + u}} \]

      16. lower-/.f64 98.2

          \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \color{blue}{\frac{t1}{t1 + u}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \frac{t1}{\color{blue}{t1 + u}} \]

      18. +-commutative N/A

          \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \frac{t1}{\color{blue}{u + t1}} \]

      19. lower-+.f64 98.2

          \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \frac{t1}{\color{blue}{u + t1}} \]

   3. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\frac{v}{\left(-t1\right) - u} \cdot \frac{t1}{u + t1}} \]

   4. Taylor expanded in u around inf

      \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \color{blue}{\frac{t1}{u}} \]
   5. Step-by-step derivation

      1. lower-/.f64 54.6

         \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \frac{t1}{\color{blue}{u}} \]

   6. Applied rewrites 54.6%

      \[\leadsto \frac{v}{\left(-t1\right) - u} \cdot \color{blue}{\frac{t1}{u}} \]

   7. Taylor expanded in u around inf

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{v}{u}\right)} \cdot \frac{t1}{u} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\frac{v}{u}}\right) \cdot \frac{t1}{u} \]

      2. lower-/.f64 52.3

         \[\leadsto \left(-1 \cdot \frac{v}{\color{blue}{u}}\right) \cdot \frac{t1}{u} \]

   9. Applied rewrites 52.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{v}{u}\right)} \cdot \frac{t1}{u} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 62.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \frac{v\_m}{\mathsf{fma}\left(-2, u, -1 \cdot t1\right)} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1)
 :precision binary64
 (* v_s (/ v_m (fma -2.0 u (* -1.0 t1)))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	return v_s * (v_m / fma(-2.0, u, (-1.0 * t1)));
}

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	return Float64(v_s * Float64(v_m / fma(-2.0, u, Float64(-1.0 * t1))))
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * N[(v$95$m / N[(-2.0 * u + N[(-1.0 * t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.6%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   2. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{\left(-t1\right) \cdot v}}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{\color{blue}{\left(-t1\right) \cdot v}}} \]

   4. associate-/r* N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}{v}}} \]

   5. div-flip-rev N/A

      \[\leadsto \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{v}{\frac{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}}{-t1}} \]

   8. sqr-neg-rev N/A

      \[\leadsto \frac{v}{\frac{\color{blue}{\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right)}}{-t1}} \]

   9. associate-/l* N/A

      \[\leadsto \frac{v}{\color{blue}{\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right) \cdot \frac{\mathsf{neg}\left(\left(t1 + u\right)\right)}{-t1}}} \]

   10. frac-2neg N/A

       \[\leadsto \frac{v}{\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right)\right)}{\mathsf{neg}\left(\left(-t1\right)\right)}}} \]

3. Applied rewrites 95.3%

   \[\leadsto \color{blue}{\frac{v}{\left(\left(-t1\right) - u\right) \cdot \left(\frac{u}{t1} - -1\right)}} \]

4. Taylor expanded in u around 0

   \[\leadsto \frac{v}{\color{blue}{-2 \cdot u + -1 \cdot t1}} \]
5. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \frac{v}{\mathsf{fma}\left(-2, \color{blue}{u}, -1 \cdot t1\right)} \]

   2. lower-*.f64 62.7

      \[\leadsto \frac{v}{\mathsf{fma}\left(-2, u, -1 \cdot t1\right)} \]

6. Applied rewrites 62.7%

   \[\leadsto \frac{v}{\color{blue}{\mathsf{fma}\left(-2, u, -1 \cdot t1\right)}} \]
7. Add Preprocessing

Alternative 7: 62.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \frac{-1 \cdot v\_m}{u + t1} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1) :precision binary64 (* v_s (/ (* -1.0 v_m) (+ u t1))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	return v_s * ((-1.0 * v_m) / (u + t1));
}

Fortran

v\_m =     private
v\_s =     private
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(v_s, u, v_m, t1)
use fmin_fmax_functions
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    code = v_s * (((-1.0d0) * v_m) / (u + t1))
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	return v_s * ((-1.0 * v_m) / (u + t1));
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	return v_s * ((-1.0 * v_m) / (u + t1))

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	return Float64(v_s * Float64(Float64(-1.0 * v_m) / Float64(u + t1)))
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp = code(v_s, u, v_m, t1)
	tmp = v_s * ((-1.0 * v_m) / (u + t1));
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * N[(N[(-1.0 * v$95$m), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.6%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(-t1\right) \cdot v}}{t1 + u}}{t1 + u} \]

   6. associate-*l/ N/A

      \[\leadsto \frac{\color{blue}{\frac{-t1}{t1 + u} \cdot v}}{t1 + u} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{-t1}{t1 + u} \cdot v}}{t1 + u} \]

   8. frac-2neg N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(\left(-t1\right)\right)}{\mathsf{neg}\left(\left(t1 + u\right)\right)}} \cdot v}{t1 + u} \]

   9. lift-neg.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)}\right)}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \cdot v}{t1 + u} \]

   10. remove-double-neg N/A

       \[\leadsto \frac{\frac{\color{blue}{t1}}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \cdot v}{t1 + u} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)}} \cdot v}{t1 + u} \]

   12. lift-+.f64 N/A

       \[\leadsto \frac{\frac{t1}{\mathsf{neg}\left(\color{blue}{\left(t1 + u\right)}\right)} \cdot v}{t1 + u} \]

   13. distribute-neg-in N/A

       \[\leadsto \frac{\frac{t1}{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) + \left(\mathsf{neg}\left(u\right)\right)}} \cdot v}{t1 + u} \]

   14. lift-neg.f64 N/A

       \[\leadsto \frac{\frac{t1}{\color{blue}{\left(-t1\right)} + \left(\mathsf{neg}\left(u\right)\right)} \cdot v}{t1 + u} \]

   15. sub-flip-reverse N/A

       \[\leadsto \frac{\frac{t1}{\color{blue}{\left(-t1\right) - u}} \cdot v}{t1 + u} \]

   16. lower--.f64 98.3

       \[\leadsto \frac{\frac{t1}{\color{blue}{\left(-t1\right) - u}} \cdot v}{t1 + u} \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{\frac{t1}{\left(-t1\right) - u} \cdot v}{\color{blue}{t1 + u}} \]

   18. +-commutative N/A

       \[\leadsto \frac{\frac{t1}{\left(-t1\right) - u} \cdot v}{\color{blue}{u + t1}} \]

   19. lower-+.f64 98.3

       \[\leadsto \frac{\frac{t1}{\left(-t1\right) - u} \cdot v}{\color{blue}{u + t1}} \]

3. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\frac{\frac{t1}{\left(-t1\right) - u} \cdot v}{u + t1}} \]

4. Taylor expanded in u around 0

   \[\leadsto \frac{\color{blue}{-1} \cdot v}{u + t1} \]
5. Step-by-step derivation

6. Applied rewrites 62.3%

   \[\leadsto \frac{\color{blue}{-1} \cdot v}{u + t1} \]
7. Add Preprocessing

Alternative 8: 62.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \left(v\_m \cdot \frac{-1}{u + t1}\right) \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1) :precision binary64 (* v_s (* v_m (/ -1.0 (+ u t1)))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	return v_s * (v_m * (-1.0 / (u + t1)));
}

Fortran

v\_m =     private
v\_s =     private
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(v_s, u, v_m, t1)
use fmin_fmax_functions
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    code = v_s * (v_m * ((-1.0d0) / (u + t1)))
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	return v_s * (v_m * (-1.0 / (u + t1)));
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	return v_s * (v_m * (-1.0 / (u + t1)))

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	return Float64(v_s * Float64(v_m * Float64(-1.0 / Float64(u + t1))))
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp = code(v_s, u, v_m, t1)
	tmp = v_s * (v_m * (-1.0 / (u + t1)));
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * N[(v$95$m * N[(-1.0 / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.6%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(-t1\right) \cdot v}}{t1 + u}}{t1 + u} \]

   6. associate-*l/ N/A

      \[\leadsto \frac{\color{blue}{\frac{-t1}{t1 + u} \cdot v}}{t1 + u} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{-t1}{t1 + u} \cdot v}}{t1 + u} \]

   8. frac-2neg N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(\left(-t1\right)\right)}{\mathsf{neg}\left(\left(t1 + u\right)\right)}} \cdot v}{t1 + u} \]

   9. lift-neg.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)}\right)}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \cdot v}{t1 + u} \]

   10. remove-double-neg N/A

       \[\leadsto \frac{\frac{\color{blue}{t1}}{\mathsf{neg}\left(\left(t1 + u\right)\right)} \cdot v}{t1 + u} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{t1}{\mathsf{neg}\left(\left(t1 + u\right)\right)}} \cdot v}{t1 + u} \]

   12. lift-+.f64 N/A

       \[\leadsto \frac{\frac{t1}{\mathsf{neg}\left(\color{blue}{\left(t1 + u\right)}\right)} \cdot v}{t1 + u} \]

   13. distribute-neg-in N/A

       \[\leadsto \frac{\frac{t1}{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) + \left(\mathsf{neg}\left(u\right)\right)}} \cdot v}{t1 + u} \]

   14. lift-neg.f64 N/A

       \[\leadsto \frac{\frac{t1}{\color{blue}{\left(-t1\right)} + \left(\mathsf{neg}\left(u\right)\right)} \cdot v}{t1 + u} \]

   15. sub-flip-reverse N/A

       \[\leadsto \frac{\frac{t1}{\color{blue}{\left(-t1\right) - u}} \cdot v}{t1 + u} \]

   16. lower--.f64 98.3

       \[\leadsto \frac{\frac{t1}{\color{blue}{\left(-t1\right) - u}} \cdot v}{t1 + u} \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{\frac{t1}{\left(-t1\right) - u} \cdot v}{\color{blue}{t1 + u}} \]

   18. +-commutative N/A

       \[\leadsto \frac{\frac{t1}{\left(-t1\right) - u} \cdot v}{\color{blue}{u + t1}} \]

   19. lower-+.f64 98.3

       \[\leadsto \frac{\frac{t1}{\left(-t1\right) - u} \cdot v}{\color{blue}{u + t1}} \]

3. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\frac{\frac{t1}{\left(-t1\right) - u} \cdot v}{u + t1}} \]

4. Taylor expanded in u around 0

   \[\leadsto \frac{\color{blue}{-1} \cdot v}{u + t1} \]
5. Step-by-step derivation

6. Applied rewrites 62.3%

   \[\leadsto \frac{\color{blue}{-1} \cdot v}{u + t1} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot v}{u + t1}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{u + t1} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{v \cdot -1}}{u + t1} \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{v \cdot -1}{\color{blue}{u + t1}} \]

   5. +-commutative N/A

      \[\leadsto \frac{v \cdot -1}{\color{blue}{t1 + u}} \]

   6. associate-/l* N/A

      \[\leadsto \color{blue}{v \cdot \frac{-1}{t1 + u}} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{v \cdot \frac{-1}{t1 + u}} \]

   8. lower-/.f64 N/A

      \[\leadsto v \cdot \color{blue}{\frac{-1}{t1 + u}} \]

   9. +-commutative N/A

      \[\leadsto v \cdot \frac{-1}{\color{blue}{u + t1}} \]

   10. lift-+.f64 62.2

       \[\leadsto v \cdot \frac{-1}{\color{blue}{u + t1}} \]

8. Applied rewrites 62.2%

   \[\leadsto \color{blue}{v \cdot \frac{-1}{u + t1}} \]
9. Add Preprocessing

Alternative 9: 54.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \frac{-1}{\frac{t1}{v\_m}} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1) :precision binary64 (* v_s (/ -1.0 (/ t1 v_m))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	return v_s * (-1.0 / (t1 / v_m));
}

Fortran

v\_m =     private
v\_s =     private
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(v_s, u, v_m, t1)
use fmin_fmax_functions
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    code = v_s * ((-1.0d0) / (t1 / v_m))
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	return v_s * (-1.0 / (t1 / v_m));
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	return v_s * (-1.0 / (t1 / v_m))

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	return Float64(v_s * Float64(-1.0 / Float64(t1 / v_m)))
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp = code(v_s, u, v_m, t1)
	tmp = v_s * (-1.0 / (t1 / v_m));
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * N[(-1.0 / N[(t1 / v$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.6%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

2. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{v}{t1}} \]

   2. lower-/.f64 54.9

      \[\leadsto -1 \cdot \frac{v}{\color{blue}{t1}} \]

4. Applied rewrites 54.9%

   \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{v}{t1}} \]

   2. lift-/.f64 N/A

      \[\leadsto -1 \cdot \frac{v}{\color{blue}{t1}} \]

   3. div-flip N/A

      \[\leadsto -1 \cdot \frac{1}{\color{blue}{\frac{t1}{v}}} \]

   4. mult-flip-rev N/A

      \[\leadsto \frac{-1}{\color{blue}{\frac{t1}{v}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{\frac{t1}{v}}} \]

   6. lower-/.f64 54.5

      \[\leadsto \frac{-1}{\frac{t1}{\color{blue}{v}}} \]

6. Applied rewrites 54.5%

   \[\leadsto \frac{-1}{\color{blue}{\frac{t1}{v}}} \]
7. Add Preprocessing

Alternative 10: 54.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \frac{-v\_m}{t1} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1) :precision binary64 (* v_s (/ (- v_m) t1)))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	return v_s * (-v_m / t1);
}

Fortran

v\_m =     private
v\_s =     private
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(v_s, u, v_m, t1)
use fmin_fmax_functions
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    code = v_s * (-v_m / t1)
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	return v_s * (-v_m / t1);
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	return v_s * (-v_m / t1)

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	return Float64(v_s * Float64(Float64(-v_m) / t1))
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp = code(v_s, u, v_m, t1)
	tmp = v_s * (-v_m / t1);
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * N[((-v$95$m) / t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.6%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

2. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{v}{t1}} \]

   2. lower-/.f64 54.9

      \[\leadsto -1 \cdot \frac{v}{\color{blue}{t1}} \]

4. Applied rewrites 54.9%

   \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{v}{t1}} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{v}{t1}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{v}{t1}\right) \]

   4. distribute-neg-frac N/A

      \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\color{blue}{t1}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\color{blue}{t1}} \]

   6. lower-neg.f64 54.9

      \[\leadsto \frac{-v}{t1} \]

6. Applied rewrites 54.9%

   \[\leadsto \frac{-v}{\color{blue}{t1}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))