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

Percentage Accurate: 61.0% → 79.6%

Time: 15.2s

Alternatives: 4

Speedup: 37.7×

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

real(8) function code(x, y, z, t, a)
    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: 61.0% 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

real(8) function code(x, y, z, t, a)
    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: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x, y, z_m, t, a] = \mathsf{sort}([x, y, z_m, t, a])\\\\ [x, y, z_m, t, a] = \mathsf{sort}([x, y, z_m, t, a])\\ \\ z\_s \cdot \left(y \cdot \frac{x}{\frac{\mathsf{fma}\left(a \cdot \frac{t}{z\_m}, -0.5, z\_m\right)}{z\_m}}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x y z_m t a)
 :precision binary64
 (* z_s (* y (/ x (/ (fma (* a (/ t z_m)) -0.5 z_m) z_m)))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x < y && y < z_m && z_m < t && t < a);
assert(x < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	return z_s * (y * (x / (fma((a * (t / z_m)), -0.5, z_m) / z_m)));
}

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
x, y, z_m, t, a = sort([x, y, z_m, t, a])
x, y, z_m, t, a = sort([x, y, z_m, t, a])
function code(z_s, x, y, z_m, t, a)
	return Float64(z_s * Float64(y * Float64(x / Float64(fma(Float64(a * Float64(t / z_m)), -0.5, z_m) / z_m))))
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]
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(y * N[(x / N[(N[(N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] * -0.5 + z$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.7%

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

   1. associate-/l* 65.7%

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

   2. associate-*l/ 65.5%

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

   3. *-commutative 65.5%

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

   4. associate-/l* 64.0%

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

3. Simplified 64.0%

   \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]
4. Add Preprocessing

5. Taylor expanded in z around inf 45.1%

   \[\leadsto y \cdot \frac{x \cdot z}{\color{blue}{z + -0.5 \cdot \frac{a \cdot t}{z}}} \]
6. Step-by-step derivation

   1. associate-/l* 45.1%

      \[\leadsto y \cdot \frac{x \cdot z}{z + -0.5 \cdot \color{blue}{\frac{a}{\frac{z}{t}}}} \]

7. Simplified 45.1%

   \[\leadsto y \cdot \frac{x \cdot z}{\color{blue}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}} \]
8. Step-by-step derivation

   1. associate-/l* 49.3%

      \[\leadsto y \cdot \color{blue}{\frac{x}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}} \]

   2. div-inv 49.3%

      \[\leadsto y \cdot \color{blue}{\left(x \cdot \frac{1}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\right)} \]

   3. +-commutative 49.3%

      \[\leadsto y \cdot \left(x \cdot \frac{1}{\frac{\color{blue}{-0.5 \cdot \frac{a}{\frac{z}{t}} + z}}{z}}\right) \]

   4. fma-def 49.3%

      \[\leadsto y \cdot \left(x \cdot \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(-0.5, \frac{a}{\frac{z}{t}}, z\right)}}{z}}\right) \]

   5. associate-/r/ 49.3%

      \[\leadsto y \cdot \left(x \cdot \frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{\frac{a}{z} \cdot t}, z\right)}{z}}\right) \]

9. Applied egg-rr 49.3%

   \[\leadsto y \cdot \color{blue}{\left(x \cdot \frac{1}{\frac{\mathsf{fma}\left(-0.5, \frac{a}{z} \cdot t, z\right)}{z}}\right)} \]
10. Step-by-step derivation

    1. un-div-inv 49.3%

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

    2. fma-udef 49.3%

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

    3. *-commutative 49.3%

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

    4. fma-def 49.3%

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

    5. associate-*l/ 49.3%

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

    6. associate-*r/ 49.3%

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

11. Applied egg-rr 49.3%

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

12. Final simplification 49.3%

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

Alternative 2: 69.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x, y, z_m, t, a] = \mathsf{sort}([x, y, z_m, t, a])\\\\ [x, y, z_m, t, a] = \mathsf{sort}([x, y, z_m, t, a])\\ \\ z\_s \cdot \left(y \cdot \frac{x \cdot z\_m}{z\_m + \left(a \cdot \frac{t}{z\_m}\right) \cdot -0.5}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x y z_m t a)
 :precision binary64
 (* z_s (* y (/ (* x z_m) (+ z_m (* (* a (/ t z_m)) -0.5))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x < y && y < z_m && z_m < t && t < a);
assert(x < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	return z_s * (y * ((x * z_m) / (z_m + ((a * (t / z_m)) * -0.5))));
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x, y, z_m, t, a)
    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 * z_m) / (z_m + ((a * (t / z_m)) * (-0.5d0)))))
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x < y && y < z_m && z_m < t && t < a;
assert x < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	return z_s * (y * ((x * z_m) / (z_m + ((a * (t / z_m)) * -0.5))));
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x, y, z_m, t, a] = sort([x, y, z_m, t, a])
[x, y, z_m, t, a] = sort([x, y, z_m, t, a])
def code(z_s, x, y, z_m, t, a):
	return z_s * (y * ((x * z_m) / (z_m + ((a * (t / z_m)) * -0.5))))

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
x, y, z_m, t, a = sort([x, y, z_m, t, a])
x, y, z_m, t, a = sort([x, y, z_m, t, a])
function code(z_s, x, y, z_m, t, a)
	return Float64(z_s * Float64(y * Float64(Float64(x * z_m) / Float64(z_m + Float64(Float64(a * Float64(t / z_m)) * -0.5)))))
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
x, y, z_m, t, a = num2cell(sort([x, y, z_m, t, a])){:}
x, y, z_m, t, a = num2cell(sort([x, y, z_m, t, a])){:}
function tmp = code(z_s, x, y, z_m, t, a)
	tmp = z_s * (y * ((x * z_m) / (z_m + ((a * (t / z_m)) * -0.5))));
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]
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(y * N[(N[(x * z$95$m), $MachinePrecision] / N[(z$95$m + N[(N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.7%

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

   1. associate-/l* 65.7%

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

   2. associate-*l/ 65.5%

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

   3. *-commutative 65.5%

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

   4. associate-/l* 64.0%

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

3. Simplified 64.0%

   \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]
4. Add Preprocessing

5. Taylor expanded in z around inf 45.1%

   \[\leadsto y \cdot \frac{x \cdot z}{\color{blue}{z + -0.5 \cdot \frac{a \cdot t}{z}}} \]
6. Step-by-step derivation

   1. associate-/l* 45.1%

      \[\leadsto y \cdot \frac{x \cdot z}{z + -0.5 \cdot \color{blue}{\frac{a}{\frac{z}{t}}}} \]

7. Simplified 45.1%

   \[\leadsto y \cdot \frac{x \cdot z}{\color{blue}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}} \]

8. Taylor expanded in a around 0 45.1%

   \[\leadsto y \cdot \frac{x \cdot z}{z + -0.5 \cdot \color{blue}{\frac{a \cdot t}{z}}} \]
9. Step-by-step derivation

   1. associate-*r/ 45.1%

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

10. Simplified 45.1%

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

11. Final simplification 45.1%

    \[\leadsto y \cdot \frac{x \cdot z}{z + \left(a \cdot \frac{t}{z}\right) \cdot -0.5} \]
12. Add Preprocessing

Alternative 3: 65.9% accurate, 16.1× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x y z_m t a) :precision binary64 (* z_s (* y (/ (* x z_m) z_m))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x < y && y < z_m && z_m < t && t < a);
assert(x < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	return z_s * (y * ((x * z_m) / z_m));
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x, y, z_m, t, a)
    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 * z_m) / z_m))
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x < y && y < z_m && z_m < t && t < a;
assert x < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	return z_s * (y * ((x * z_m) / z_m));
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x, y, z_m, t, a] = sort([x, y, z_m, t, a])
[x, y, z_m, t, a] = sort([x, y, z_m, t, a])
def code(z_s, x, y, z_m, t, a):
	return z_s * (y * ((x * z_m) / z_m))

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
x, y, z_m, t, a = sort([x, y, z_m, t, a])
x, y, z_m, t, a = sort([x, y, z_m, t, a])
function code(z_s, x, y, z_m, t, a)
	return Float64(z_s * Float64(y * Float64(Float64(x * z_m) / z_m)))
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
x, y, z_m, t, a = num2cell(sort([x, y, z_m, t, a])){:}
x, y, z_m, t, a = num2cell(sort([x, y, z_m, t, a])){:}
function tmp = code(z_s, x, y, z_m, t, a)
	tmp = z_s * (y * ((x * z_m) / z_m));
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]
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(y * N[(N[(x * z$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.7%

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

   1. associate-/l* 65.7%

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

   2. associate-*l/ 65.5%

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

   3. *-commutative 65.5%

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

   4. associate-/l* 64.0%

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

3. Simplified 64.0%

   \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]
4. Add Preprocessing

5. Taylor expanded in z around inf 40.4%

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

6. Final simplification 40.4%

   \[\leadsto y \cdot \frac{x \cdot z}{z} \]
7. Add Preprocessing

Alternative 4: 73.4% accurate, 37.7× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
(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);
assert(x < y && y < z_m && z_m < t && t < a);
assert(x < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	return z_s * (y * x);
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x, y, z_m, t, a)
    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);
assert x < y && y < z_m && z_m < t && t < a;
assert x < y && y < z_m && z_m < t && t < a;
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)
[x, y, z_m, t, a] = sort([x, y, z_m, t, a])
[x, y, z_m, t, a] = sort([x, y, z_m, t, a])
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)
x, y, z_m, t, a = sort([x, y, z_m, t, a])
x, y, z_m, t, a = sort([x, y, z_m, t, a])
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);
x, y, z_m, t, a = num2cell(sort([x, y, z_m, t, a])){:}
x, y, z_m, t, a = num2cell(sort([x, y, z_m, t, a])){:}
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]
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(y * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.7%

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

   1. associate-/l* 65.7%

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

   2. associate-*l/ 65.5%

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

   3. *-commutative 65.5%

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

   4. associate-/l* 64.0%

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

3. Simplified 64.0%

   \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]
4. Add Preprocessing

5. Taylor expanded in z around inf 41.7%

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

6. Final simplification 41.7%

   \[\leadsto y \cdot x \]
7. Add Preprocessing

Developer target: 89.7% accurate, 0.9× 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

real(8) function code(x, y, z, t, a)
    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 2024033 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

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

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