Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2

Percentage Accurate: 60.5% → 90.3%

Time: 9.8s

Alternatives: 8

Speedup: 7.5×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / sqrt(((z * z) - (t * a)));
}

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(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((x * y) * z) / sqrt(((z * z) - (t * a)))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / Math.sqrt(((z * z) - (t * a)));
}

Python

def code(x, y, z, t, a):
	return ((x * y) * z) / math.sqrt(((z * z) - (t * a)))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) * z) / sqrt(Float64(Float64(z * z) - Float64(t * a))))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) * z) / sqrt(((z * z) - (t * a)));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 60.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / sqrt(((z * z) - (t * a)));
}

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(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((x * y) * z) / sqrt(((z * z) - (t * a)))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / Math.sqrt(((z * z) - (t * a)));
}

Python

def code(x, y, z, t, a):
	return ((x * y) * z) / math.sqrt(((z * z) - (t * a)))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) * z) / sqrt(Float64(Float64(z * z) - Float64(t * a))))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) * z) / sqrt(((z * z) - (t * a)));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 90.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.5 \cdot 10^{+65}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z\_m}{\sqrt{\mathsf{fma}\left(-t, a, z\_m \cdot z\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z\_m}{\mathsf{fma}\left(a \cdot \frac{t}{z\_m}, -0.5, z\_m\right)}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (*
  z_s
  (if (<= z_m 3.5e+65)
    (* (* y x) (/ z_m (sqrt (fma (- t) a (* z_m z_m)))))
    (* (* y x) (/ z_m (fma (* a (/ t z_m)) -0.5 z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.5e+65) {
		tmp = (y * x) * (z_m / sqrt(fma(-t, a, (z_m * z_m))));
	} else {
		tmp = (y * x) * (z_m / fma((a * (t / z_m)), -0.5, z_m));
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3.5e+65)
		tmp = Float64(Float64(y * x) * Float64(z_m / sqrt(fma(Float64(-t), a, Float64(z_m * z_m)))));
	else
		tmp = Float64(Float64(y * x) * Float64(z_m / fma(Float64(a * Float64(t / z_m)), -0.5, z_m)));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 3.5e+65], N[(N[(y * x), $MachinePrecision] * N[(z$95$m / N[Sqrt[N[((-t) * a + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] * N[(z$95$m / N[(N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] * -0.5 + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.5000000000000001e65

   1. Initial program 63.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right)} \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot z}}{\sqrt{z \cdot z - t \cdot a}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\sqrt{z \cdot z - t \cdot a}}} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{z \cdot z - t \cdot a}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{z \cdot z} - t \cdot a}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - \color{blue}{t \cdot a}}} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \color{blue}{\frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\color{blue}{\sqrt{z \cdot z - t \cdot a}}} \]

      14. pow2 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{{z}^{2}} - t \cdot a}} \]

      15. *-commutative N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{{z}^{2} - \color{blue}{a \cdot t}}} \]

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

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{{z}^{2} + \left(\mathsf{neg}\left(a\right)\right) \cdot t}}} \]

      17. mul-1-neg N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{{z}^{2} + \color{blue}{\left(-1 \cdot a\right)} \cdot t}} \]

      18. associate-*r* N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{{z}^{2} + \color{blue}{-1 \cdot \left(a \cdot t\right)}}} \]

      19. +-commutative N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{-1 \cdot \left(a \cdot t\right) + {z}^{2}}}} \]

      20. associate-*r* N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot t} + {z}^{2}}} \]

      21. mul-1-neg N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot t + {z}^{2}}} \]

      22. lower-fma.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), t, {z}^{2}\right)}}} \]

      23. lower-neg.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(\color{blue}{-a}, t, {z}^{2}\right)}} \]

      24. pow2 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, \color{blue}{z \cdot z}\right)}} \]

      25. lift-*.f64 65.7

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, \color{blue}{z \cdot z}\right)}} \]

   4. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{-1 \cdot \left(a \cdot t\right)}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}} \]

      2. distribute-lft-neg-out N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot \color{blue}{t}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot \color{blue}{t}}} \]

      4. lift-neg.f64 39.7

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(-a\right) \cdot t}} \]

   7. Applied rewrites 39.7%

      \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{\left(-a\right) \cdot t}}} \]

   8. Taylor expanded in z around 0

      \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{-1 \cdot \left(a \cdot t\right) + {z}^{2}}}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(a \cdot t\right) \cdot -1 + {\color{blue}{z}}^{2}}} \]

      2. associate-*r* N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{a \cdot \left(t \cdot -1\right) + {\color{blue}{z}}^{2}}} \]

      3. metadata-eval N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{a \cdot \left(t \cdot {-1}^{1}\right) + {z}^{2}}} \]

      4. metadata-eval N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{a \cdot \left(t \cdot {-1}^{\left(\frac{2}{2}\right)}\right) + {z}^{2}}} \]

      5. sqrt-pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{a \cdot \left(t \cdot {\left(\sqrt{-1}\right)}^{2}\right) + {z}^{2}}} \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(t \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot a + {\color{blue}{z}}^{2}}} \]

      7. sqrt-pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(t \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot a + {z}^{2}}} \]

      8. metadata-eval N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(t \cdot {-1}^{1}\right) \cdot a + {z}^{2}}} \]

      9. metadata-eval N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(t \cdot -1\right) \cdot a + {z}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(-1 \cdot t\right) \cdot a + {z}^{2}}} \]

      11. mul-1-neg N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(\mathsf{neg}\left(t\right)\right) \cdot a + {z}^{2}}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(\mathsf{neg}\left(t\right), \color{blue}{a}, {z}^{2}\right)}} \]

      13. lower-neg.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-t, a, {z}^{2}\right)}} \]

      14. pow2 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-t, a, z \cdot z\right)}} \]

      15. lift-*.f64 65.7

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-t, a, z \cdot z\right)}} \]

   10. Applied rewrites 65.7%

       \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{\mathsf{fma}\left(-t, a, z \cdot z\right)}}} \]

   if 3.5000000000000001e65 < z

   1. Initial program 42.9%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z + \frac{-1}{2} \cdot \frac{a \cdot t}{z}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\frac{-1}{2} \cdot \frac{a \cdot t}{z} + \color{blue}{z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\frac{a \cdot t}{z} \cdot \frac{-1}{2} + z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \color{blue}{\frac{-1}{2}}, z\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)} \]

      5. lower-*.f64 80.4

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, -0.5, z\right)} \]

   5. Applied rewrites 80.4%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\mathsf{fma}\left(\frac{a \cdot t}{z}, -0.5, z\right)}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot z}}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right)} \cdot z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)} \]

      8. lower-/.f64 94.4

         \[\leadsto \left(y \cdot x\right) \cdot \color{blue}{\frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, -0.5, z\right)}} \]

   7. Applied rewrites 94.4%

      \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, -0.5, z\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)} \]

      3. associate-/l* N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(a \cdot \frac{t}{z}, \frac{-1}{2}, z\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(a \cdot \frac{t}{z}, \frac{-1}{2}, z\right)} \]

      5. lower-/.f64 95.8

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(a \cdot \frac{t}{z}, -0.5, z\right)} \]

   9. Applied rewrites 95.8%

      \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(a \cdot \frac{t}{z}, -0.5, z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 71.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot y\right) \cdot z\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}} \leq 2 \cdot 10^{-222}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot z\_m\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (*
  z_s
  (if (<= (/ (* (* x y) z_m) (sqrt (- (* z_m z_m) (* t a)))) 2e-222)
    (/ (* y (* x z_m)) z_m)
    (* y x))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if ((((x * y) * z_m) / sqrt(((z_m * z_m) - (t * a)))) <= 2e-222) {
		tmp = (y * (x * z_m)) / z_m;
	} else {
		tmp = y * x;
	}
	return z_s * tmp;
}

Fortran

z\_m =     private
z\_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(z_s, x, y, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((((x * y) * z_m) / sqrt(((z_m * z_m) - (t * a)))) <= 2d-222) then
        tmp = (y * (x * z_m)) / z_m
    else
        tmp = y * x
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if ((((x * y) * z_m) / Math.sqrt(((z_m * z_m) - (t * a)))) <= 2e-222) {
		tmp = (y * (x * z_m)) / z_m;
	} else {
		tmp = y * x;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t, a):
	tmp = 0
	if (((x * y) * z_m) / math.sqrt(((z_m * z_m) - (t * a)))) <= 2e-222:
		tmp = (y * (x * z_m)) / z_m
	else:
		tmp = y * x
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (Float64(Float64(Float64(x * y) * z_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))) <= 2e-222)
		tmp = Float64(Float64(y * Float64(x * z_m)) / z_m);
	else
		tmp = Float64(y * x);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t, a)
	tmp = 0.0;
	if ((((x * y) * z_m) / sqrt(((z_m * z_m) - (t * a)))) <= 2e-222)
		tmp = (y * (x * z_m)) / z_m;
	else
		tmp = y * x;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[N[(N[(N[(x * y), $MachinePrecision] * z$95$m), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-222], N[(N[(y * N[(x * z$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y * x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 2.0000000000000001e-222

   1. Initial program 66.6%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z}} \]
   4. Step-by-step derivation

   5. Applied rewrites 50.6%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right)} \cdot z}{z} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot z}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot x\right)} \cdot z}{z} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot z\right)}}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot z\right)}}{z} \]

      6. lower-*.f64 49.4

         \[\leadsto \frac{y \cdot \color{blue}{\left(x \cdot z\right)}}{z} \]

   7. Applied rewrites 49.4%

      \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot z\right)}}{z} \]

   if 2.0000000000000001e-222 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   1. Initial program 41.5%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{x} \]

      2. lower-*.f64 40.3

         \[\leadsto y \cdot \color{blue}{x} \]

   5. Applied rewrites 40.3%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 71.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot y\right) \cdot z\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}} \leq 2 \cdot 10^{-222}:\\ \;\;\;\;y \cdot \frac{z\_m \cdot x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (*
  z_s
  (if (<= (/ (* (* x y) z_m) (sqrt (- (* z_m z_m) (* t a)))) 2e-222)
    (* y (/ (* z_m x) z_m))
    (* y x))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if ((((x * y) * z_m) / sqrt(((z_m * z_m) - (t * a)))) <= 2e-222) {
		tmp = y * ((z_m * x) / z_m);
	} else {
		tmp = y * x;
	}
	return z_s * tmp;
}

Fortran

z\_m =     private
z\_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(z_s, x, y, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((((x * y) * z_m) / sqrt(((z_m * z_m) - (t * a)))) <= 2d-222) then
        tmp = y * ((z_m * x) / z_m)
    else
        tmp = y * x
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if ((((x * y) * z_m) / Math.sqrt(((z_m * z_m) - (t * a)))) <= 2e-222) {
		tmp = y * ((z_m * x) / z_m);
	} else {
		tmp = y * x;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t, a):
	tmp = 0
	if (((x * y) * z_m) / math.sqrt(((z_m * z_m) - (t * a)))) <= 2e-222:
		tmp = y * ((z_m * x) / z_m)
	else:
		tmp = y * x
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (Float64(Float64(Float64(x * y) * z_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))) <= 2e-222)
		tmp = Float64(y * Float64(Float64(z_m * x) / z_m));
	else
		tmp = Float64(y * x);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t, a)
	tmp = 0.0;
	if ((((x * y) * z_m) / sqrt(((z_m * z_m) - (t * a)))) <= 2e-222)
		tmp = y * ((z_m * x) / z_m);
	else
		tmp = y * x;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[N[(N[(N[(x * y), $MachinePrecision] * z$95$m), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-222], N[(y * N[(N[(z$95$m * x), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 2.0000000000000001e-222

   1. Initial program 66.6%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z}} \]
   4. Step-by-step derivation

   5. Applied rewrites 50.6%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right)} \cdot z}{z} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot z}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot x\right)} \cdot z}{z} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot z\right)}}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot z\right)}}{z} \]

      6. lower-*.f64 49.4

         \[\leadsto \frac{y \cdot \color{blue}{\left(x \cdot z\right)}}{z} \]

   7. Applied rewrites 49.4%

      \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot z\right)}}{z} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(x \cdot z\right)}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot z\right)}}{z} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(x \cdot z\right)}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{z}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{z}} \]

      6. lower-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{x \cdot z}{z}} \]

      7. *-commutative N/A

         \[\leadsto y \cdot \frac{\color{blue}{z \cdot x}}{z} \]

      8. lower-*.f64 45.8

         \[\leadsto y \cdot \frac{\color{blue}{z \cdot x}}{z} \]

   9. Applied rewrites 45.8%

      \[\leadsto \color{blue}{y \cdot \frac{z \cdot x}{z}} \]

   if 2.0000000000000001e-222 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   1. Initial program 41.5%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{x} \]

      2. lower-*.f64 40.3

         \[\leadsto y \cdot \color{blue}{x} \]

   5. Applied rewrites 40.3%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 90.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.4 \cdot 10^{+65}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z\_m}{\mathsf{fma}\left(a \cdot \frac{t}{z\_m}, -0.5, z\_m\right)}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (*
  z_s
  (if (<= z_m 3.4e+65)
    (* y (* x (/ z_m (sqrt (fma (- a) t (* z_m z_m))))))
    (* (* y x) (/ z_m (fma (* a (/ t z_m)) -0.5 z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.4e+65) {
		tmp = y * (x * (z_m / sqrt(fma(-a, t, (z_m * z_m)))));
	} else {
		tmp = (y * x) * (z_m / fma((a * (t / z_m)), -0.5, z_m));
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3.4e+65)
		tmp = Float64(y * Float64(x * Float64(z_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m))))));
	else
		tmp = Float64(Float64(y * x) * Float64(z_m / fma(Float64(a * Float64(t / z_m)), -0.5, z_m)));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 3.4e+65], N[(y * N[(x * N[(z$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] * N[(z$95$m / N[(N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] * -0.5 + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.3999999999999999e65

   1. Initial program 63.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right)} \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot z}}{\sqrt{z \cdot z - t \cdot a}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\sqrt{z \cdot z - t \cdot a}}} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{z \cdot z - t \cdot a}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{z \cdot z} - t \cdot a}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - \color{blue}{t \cdot a}}} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \color{blue}{\frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\color{blue}{\sqrt{z \cdot z - t \cdot a}}} \]

      14. pow2 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{{z}^{2}} - t \cdot a}} \]

      15. *-commutative N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{{z}^{2} - \color{blue}{a \cdot t}}} \]

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

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{{z}^{2} + \left(\mathsf{neg}\left(a\right)\right) \cdot t}}} \]

      17. mul-1-neg N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{{z}^{2} + \color{blue}{\left(-1 \cdot a\right)} \cdot t}} \]

      18. associate-*r* N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{{z}^{2} + \color{blue}{-1 \cdot \left(a \cdot t\right)}}} \]

      19. +-commutative N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{-1 \cdot \left(a \cdot t\right) + {z}^{2}}}} \]

      20. associate-*r* N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot t} + {z}^{2}}} \]

      21. mul-1-neg N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot t + {z}^{2}}} \]

      22. lower-fma.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), t, {z}^{2}\right)}}} \]

      23. lower-neg.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(\color{blue}{-a}, t, {z}^{2}\right)}} \]

      24. pow2 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, \color{blue}{z \cdot z}\right)}} \]

      25. lift-*.f64 65.7

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, \color{blue}{z \cdot z}\right)}} \]

   4. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \color{blue}{\frac{z}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\color{blue}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, z \cdot z\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(\mathsf{neg}\left(a\right), t, \color{blue}{z \cdot z}\right)}} \]

      7. lift-fma.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot t + z \cdot z}}} \]

      8. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{z}{\sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot t + z \cdot z}}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{z}{\sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot t + z \cdot z}}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(x \cdot \frac{z}{\sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot t + z \cdot z}}\right)} \]

      11. lift-fma.f64 N/A

          \[\leadsto y \cdot \left(x \cdot \frac{z}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), t, z \cdot z\right)}}}\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto y \cdot \left(x \cdot \frac{z}{\sqrt{\mathsf{fma}\left(\color{blue}{-a}, t, z \cdot z\right)}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto y \cdot \left(x \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, \color{blue}{z \cdot z}\right)}}\right) \]

      14. lift-sqrt.f64 N/A

          \[\leadsto y \cdot \left(x \cdot \frac{z}{\color{blue}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}}}\right) \]

      15. lift-/.f64 66.4

          \[\leadsto y \cdot \left(x \cdot \color{blue}{\frac{z}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}}}\right) \]

   6. Applied rewrites 66.4%

      \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}}\right)} \]

   if 3.3999999999999999e65 < z

   1. Initial program 42.9%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z + \frac{-1}{2} \cdot \frac{a \cdot t}{z}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\frac{-1}{2} \cdot \frac{a \cdot t}{z} + \color{blue}{z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\frac{a \cdot t}{z} \cdot \frac{-1}{2} + z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \color{blue}{\frac{-1}{2}}, z\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)} \]

      5. lower-*.f64 80.4

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, -0.5, z\right)} \]

   5. Applied rewrites 80.4%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\mathsf{fma}\left(\frac{a \cdot t}{z}, -0.5, z\right)}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot z}}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right)} \cdot z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)} \]

      8. lower-/.f64 94.4

         \[\leadsto \left(y \cdot x\right) \cdot \color{blue}{\frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, -0.5, z\right)}} \]

   7. Applied rewrites 94.4%

      \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, -0.5, z\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)} \]

      3. associate-/l* N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(a \cdot \frac{t}{z}, \frac{-1}{2}, z\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(a \cdot \frac{t}{z}, \frac{-1}{2}, z\right)} \]

      5. lower-/.f64 95.8

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(a \cdot \frac{t}{z}, -0.5, z\right)} \]

   9. Applied rewrites 95.8%

      \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(a \cdot \frac{t}{z}, -0.5, z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.55 \cdot 10^{-118}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z\_m}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z\_m}{\mathsf{fma}\left(a \cdot \frac{t}{z\_m}, -0.5, z\_m\right)}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (*
  z_s
  (if (<= z_m 1.55e-118)
    (* (* y x) (/ z_m (sqrt (* (- a) t))))
    (* (* y x) (/ z_m (fma (* a (/ t z_m)) -0.5 z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.55e-118) {
		tmp = (y * x) * (z_m / sqrt((-a * t)));
	} else {
		tmp = (y * x) * (z_m / fma((a * (t / z_m)), -0.5, z_m));
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.55e-118)
		tmp = Float64(Float64(y * x) * Float64(z_m / sqrt(Float64(Float64(-a) * t))));
	else
		tmp = Float64(Float64(y * x) * Float64(z_m / fma(Float64(a * Float64(t / z_m)), -0.5, z_m)));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 1.55e-118], N[(N[(y * x), $MachinePrecision] * N[(z$95$m / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] * N[(z$95$m / N[(N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] * -0.5 + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.5500000000000001e-118

   1. Initial program 58.7%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right)} \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot z}}{\sqrt{z \cdot z - t \cdot a}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\sqrt{z \cdot z - t \cdot a}}} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{z \cdot z - t \cdot a}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{z \cdot z} - t \cdot a}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - \color{blue}{t \cdot a}}} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \color{blue}{\frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\color{blue}{\sqrt{z \cdot z - t \cdot a}}} \]

      14. pow2 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{{z}^{2}} - t \cdot a}} \]

      15. *-commutative N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{{z}^{2} - \color{blue}{a \cdot t}}} \]

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

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{{z}^{2} + \left(\mathsf{neg}\left(a\right)\right) \cdot t}}} \]

      17. mul-1-neg N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{{z}^{2} + \color{blue}{\left(-1 \cdot a\right)} \cdot t}} \]

      18. associate-*r* N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{{z}^{2} + \color{blue}{-1 \cdot \left(a \cdot t\right)}}} \]

      19. +-commutative N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{-1 \cdot \left(a \cdot t\right) + {z}^{2}}}} \]

      20. associate-*r* N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot t} + {z}^{2}}} \]

      21. mul-1-neg N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot t + {z}^{2}}} \]

      22. lower-fma.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), t, {z}^{2}\right)}}} \]

      23. lower-neg.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(\color{blue}{-a}, t, {z}^{2}\right)}} \]

      24. pow2 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, \color{blue}{z \cdot z}\right)}} \]

      25. lift-*.f64 60.9

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, \color{blue}{z \cdot z}\right)}} \]

   4. Applied rewrites 60.9%

      \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{-1 \cdot \left(a \cdot t\right)}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}} \]

      2. distribute-lft-neg-out N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot \color{blue}{t}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot \color{blue}{t}}} \]

      4. lift-neg.f64 43.5

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(-a\right) \cdot t}} \]

   7. Applied rewrites 43.5%

      \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{\left(-a\right) \cdot t}}} \]

   if 1.5500000000000001e-118 < z

   1. Initial program 56.3%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z + \frac{-1}{2} \cdot \frac{a \cdot t}{z}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\frac{-1}{2} \cdot \frac{a \cdot t}{z} + \color{blue}{z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\frac{a \cdot t}{z} \cdot \frac{-1}{2} + z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \color{blue}{\frac{-1}{2}}, z\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)} \]

      5. lower-*.f64 78.8

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, -0.5, z\right)} \]

   5. Applied rewrites 78.8%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\mathsf{fma}\left(\frac{a \cdot t}{z}, -0.5, z\right)}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot z}}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right)} \cdot z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)} \]

      8. lower-/.f64 89.1

         \[\leadsto \left(y \cdot x\right) \cdot \color{blue}{\frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, -0.5, z\right)}} \]

   7. Applied rewrites 89.1%

      \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, -0.5, z\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(\frac{a \cdot t}{z}, \frac{-1}{2}, z\right)} \]

      3. associate-/l* N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(a \cdot \frac{t}{z}, \frac{-1}{2}, z\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(a \cdot \frac{t}{z}, \frac{-1}{2}, z\right)} \]

      5. lower-/.f64 90.1

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(a \cdot \frac{t}{z}, -0.5, z\right)} \]

   9. Applied rewrites 90.1%

      \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(a \cdot \frac{t}{z}, -0.5, z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6.6 \cdot 10^{-111}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z\_m}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (* z_s (if (<= z_m 6.6e-111) (* (* y x) (/ z_m (sqrt (* (- a) t)))) (* y x))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6.6e-111) {
		tmp = (y * x) * (z_m / sqrt((-a * t)));
	} else {
		tmp = y * x;
	}
	return z_s * tmp;
}

Fortran

z\_m =     private
z\_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(z_s, x, y, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 6.6d-111) then
        tmp = (y * x) * (z_m / sqrt((-a * t)))
    else
        tmp = y * x
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6.6e-111) {
		tmp = (y * x) * (z_m / Math.sqrt((-a * t)));
	} else {
		tmp = y * x;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t, a):
	tmp = 0
	if z_m <= 6.6e-111:
		tmp = (y * x) * (z_m / math.sqrt((-a * t)))
	else:
		tmp = y * x
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 6.6e-111)
		tmp = Float64(Float64(y * x) * Float64(z_m / sqrt(Float64(Float64(-a) * t))));
	else
		tmp = Float64(y * x);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 6.6e-111)
		tmp = (y * x) * (z_m / sqrt((-a * t)));
	else
		tmp = y * x;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 6.6e-111], N[(N[(y * x), $MachinePrecision] * N[(z$95$m / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.6e-111

   1. Initial program 58.7%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right)} \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot z}}{\sqrt{z \cdot z - t \cdot a}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\sqrt{z \cdot z - t \cdot a}}} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{z \cdot z - t \cdot a}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{z \cdot z} - t \cdot a}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - \color{blue}{t \cdot a}}} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \color{blue}{\frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\color{blue}{\sqrt{z \cdot z - t \cdot a}}} \]

      14. pow2 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{{z}^{2}} - t \cdot a}} \]

      15. *-commutative N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{{z}^{2} - \color{blue}{a \cdot t}}} \]

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

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{{z}^{2} + \left(\mathsf{neg}\left(a\right)\right) \cdot t}}} \]

      17. mul-1-neg N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{{z}^{2} + \color{blue}{\left(-1 \cdot a\right)} \cdot t}} \]

      18. associate-*r* N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{{z}^{2} + \color{blue}{-1 \cdot \left(a \cdot t\right)}}} \]

      19. +-commutative N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{-1 \cdot \left(a \cdot t\right) + {z}^{2}}}} \]

      20. associate-*r* N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot t} + {z}^{2}}} \]

      21. mul-1-neg N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot t + {z}^{2}}} \]

      22. lower-fma.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), t, {z}^{2}\right)}}} \]

      23. lower-neg.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(\color{blue}{-a}, t, {z}^{2}\right)}} \]

      24. pow2 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, \color{blue}{z \cdot z}\right)}} \]

      25. lift-*.f64 60.9

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, \color{blue}{z \cdot z}\right)}} \]

   4. Applied rewrites 60.9%

      \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{-1 \cdot \left(a \cdot t\right)}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}} \]

      2. distribute-lft-neg-out N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot \color{blue}{t}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot \color{blue}{t}}} \]

      4. lift-neg.f64 43.5

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(-a\right) \cdot t}} \]

   7. Applied rewrites 43.5%

      \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{\left(-a\right) \cdot t}}} \]

   if 6.6e-111 < z

   1. Initial program 56.3%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{x} \]

      2. lower-*.f64 88.6

         \[\leadsto y \cdot \color{blue}{x} \]

   5. Applied rewrites 88.6%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6.6 \cdot 10^{-111}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z\_m}{\sqrt{\left(-a\right) \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (* z_s (if (<= z_m 6.6e-111) (* y (* x (/ z_m (sqrt (* (- a) t))))) (* y x))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6.6e-111) {
		tmp = y * (x * (z_m / sqrt((-a * t))));
	} else {
		tmp = y * x;
	}
	return z_s * tmp;
}

Fortran

z\_m =     private
z\_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(z_s, x, y, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 6.6d-111) then
        tmp = y * (x * (z_m / sqrt((-a * t))))
    else
        tmp = y * x
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6.6e-111) {
		tmp = y * (x * (z_m / Math.sqrt((-a * t))));
	} else {
		tmp = y * x;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t, a):
	tmp = 0
	if z_m <= 6.6e-111:
		tmp = y * (x * (z_m / math.sqrt((-a * t))))
	else:
		tmp = y * x
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 6.6e-111)
		tmp = Float64(y * Float64(x * Float64(z_m / sqrt(Float64(Float64(-a) * t)))));
	else
		tmp = Float64(y * x);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 6.6e-111)
		tmp = y * (x * (z_m / sqrt((-a * t))));
	else
		tmp = y * x;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 6.6e-111], N[(y * N[(x * N[(z$95$m / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.6e-111

   1. Initial program 58.7%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right)} \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot z}}{\sqrt{z \cdot z - t \cdot a}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\sqrt{z \cdot z - t \cdot a}}} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{z \cdot z - t \cdot a}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{z \cdot z} - t \cdot a}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - \color{blue}{t \cdot a}}} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \color{blue}{\frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\color{blue}{\sqrt{z \cdot z - t \cdot a}}} \]

      14. pow2 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{{z}^{2}} - t \cdot a}} \]

      15. *-commutative N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{{z}^{2} - \color{blue}{a \cdot t}}} \]

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

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{{z}^{2} + \left(\mathsf{neg}\left(a\right)\right) \cdot t}}} \]

      17. mul-1-neg N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{{z}^{2} + \color{blue}{\left(-1 \cdot a\right)} \cdot t}} \]

      18. associate-*r* N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{{z}^{2} + \color{blue}{-1 \cdot \left(a \cdot t\right)}}} \]

      19. +-commutative N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{-1 \cdot \left(a \cdot t\right) + {z}^{2}}}} \]

      20. associate-*r* N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot t} + {z}^{2}}} \]

      21. mul-1-neg N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot t + {z}^{2}}} \]

      22. lower-fma.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), t, {z}^{2}\right)}}} \]

      23. lower-neg.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(\color{blue}{-a}, t, {z}^{2}\right)}} \]

      24. pow2 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, \color{blue}{z \cdot z}\right)}} \]

      25. lift-*.f64 60.9

          \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, \color{blue}{z \cdot z}\right)}} \]

   4. Applied rewrites 60.9%

      \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{-1 \cdot \left(a \cdot t\right)}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}} \]

      2. distribute-lft-neg-out N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot \color{blue}{t}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot \color{blue}{t}}} \]

      4. lift-neg.f64 43.5

         \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(-a\right) \cdot t}} \]

   7. Applied rewrites 43.5%

      \[\leadsto \left(y \cdot x\right) \cdot \frac{z}{\sqrt{\color{blue}{\left(-a\right) \cdot t}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(-a\right) \cdot t}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{z}{\sqrt{\left(-a\right) \cdot t}} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{z}{\sqrt{\left(-a\right) \cdot t}}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{z}{\sqrt{\left(-a\right) \cdot t}}\right)} \]

      5. lower-*.f64 41.9

         \[\leadsto y \cdot \color{blue}{\left(x \cdot \frac{z}{\sqrt{\left(-a\right) \cdot t}}\right)} \]

   9. Applied rewrites 41.9%

      \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{z}{\sqrt{\left(-a\right) \cdot t}}\right)} \]

   if 6.6e-111 < z

   1. Initial program 56.3%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{x} \]

      2. lower-*.f64 88.6

         \[\leadsto y \cdot \color{blue}{x} \]

   5. Applied rewrites 88.6%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 72.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y \cdot x\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a) :precision binary64 (* z_s (* y x)))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	return z_s * (y * x);
}

Fortran

z\_m =     private
z\_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(z_s, x, y, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = z_s * (y * x)
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	return z_s * (y * x);
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t, a):
	return z_s * (y * x)

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	return Float64(z_s * Float64(y * x))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m, t, a)
	tmp = z_s * (y * x);
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(y * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.7%

   \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{x \cdot y} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto y \cdot \color{blue}{x} \]

   2. lower-*.f64 43.2

      \[\leadsto y \cdot \color{blue}{x} \]

5. Applied rewrites 43.2%

   \[\leadsto \color{blue}{y \cdot x} \]
6. Add Preprocessing

Developer Target 1: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (< z -3.1921305903852764e+46)
   (- (* y x))
   (if (< z 5.976268120920894e+90)
     (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y))
     (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z < -3.1921305903852764e+46) {
		tmp = -(y * x);
	} else if (z < 5.976268120920894e+90) {
		tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z < (-3.1921305903852764d+46)) then
        tmp = -(y * x)
    else if (z < 5.976268120920894d+90) then
        tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z < -3.1921305903852764e+46) {
		tmp = -(y * x);
	} else if (z < 5.976268120920894e+90) {
		tmp = (x * z) / (Math.sqrt(((z * z) - (a * t))) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z < -3.1921305903852764e+46:
		tmp = -(y * x)
	elif z < 5.976268120920894e+90:
		tmp = (x * z) / (math.sqrt(((z * z) - (a * t))) / y)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z < -3.1921305903852764e+46)
		tmp = Float64(-Float64(y * x));
	elseif (z < 5.976268120920894e+90)
		tmp = Float64(Float64(x * z) / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / y));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z < -3.1921305903852764e+46)
		tmp = -(y * x);
	elseif (z < 5.976268120920894e+90)
		tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Less[z, -3.1921305903852764e+46], (-N[(y * x), $MachinePrecision]), If[Less[z, 5.976268120920894e+90], N[(N[(x * z), $MachinePrecision] / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Reproduce

herbie shell --seed 2025043 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -31921305903852764000000000000000000000000000000) (- (* y x)) (if (< z 5976268120920894000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y)) (* y x))))

  (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))