Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3

Percentage Accurate: 84.8% → 96.7%

Time: 9.3s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y, z, t):
	return (x * (y - z)) / (t - z)

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y, z, t):
	return (x * (y - z)) / (t - z)

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\frac{x\_m \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z - t} \cdot \left(z - y\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2e-26) (/ (* x_m (- y z)) (- t z)) (* (/ x_m (- z t)) (- z y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 2e-26) {
		tmp = (x_m * (y - z)) / (t - z);
	} else {
		tmp = (x_m / (z - t)) * (z - y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x_m <= 2d-26) then
        tmp = (x_m * (y - z)) / (t - z)
    else
        tmp = (x_m / (z - t)) * (z - y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 2e-26) {
		tmp = (x_m * (y - z)) / (t - z);
	} else {
		tmp = (x_m / (z - t)) * (z - y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if x_m <= 2e-26:
		tmp = (x_m * (y - z)) / (t - z)
	else:
		tmp = (x_m / (z - t)) * (z - y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 2e-26)
		tmp = Float64(Float64(x_m * Float64(y - z)) / Float64(t - z));
	else
		tmp = Float64(Float64(x_m / Float64(z - t)) * Float64(z - y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (x_m <= 2e-26)
		tmp = (x_m * (y - z)) / (t - z);
	else
		tmp = (x_m / (z - t)) * (z - y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[x$95$m, 2e-26], N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(z - t), $MachinePrecision]), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-26

   1. Initial program 88.1%

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

   if 2.0000000000000001e-26 < x

   1. Initial program 65.4%

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

      1. associate-/l* 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in x around 0 65.4%

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

      1. remove-double-neg 65.4%

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

      2. distribute-neg-frac2 65.4%

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

      3. *-commutative 65.4%

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

      4. associate-/l* 98.2%

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

      5. distribute-lft-neg-out 98.2%

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

      6. neg-sub0 98.2%

         \[\leadsto \color{blue}{\left(0 - \left(y - z\right)\right)} \cdot \frac{x}{-\left(t - z\right)} \]

      7. associate--r- 98.2%

         \[\leadsto \color{blue}{\left(\left(0 - y\right) + z\right)} \cdot \frac{x}{-\left(t - z\right)} \]

      8. neg-sub0 98.2%

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

      9. +-commutative 98.2%

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

      10. sub-neg 98.2%

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

      11. neg-sub0 98.2%

          \[\leadsto \left(z - y\right) \cdot \frac{x}{\color{blue}{0 - \left(t - z\right)}} \]

      12. associate--r- 98.2%

          \[\leadsto \left(z - y\right) \cdot \frac{x}{\color{blue}{\left(0 - t\right) + z}} \]

      13. neg-sub0 98.2%

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

      14. +-commutative 98.2%

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

      15. sub-neg 98.2%

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

      16. *-commutative 98.2%

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

   7. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{x}{z - t} \cdot \left(z - y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - t} \cdot \left(z - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{y - z}{t}\\ t_2 := x\_m \cdot \left(1 - \frac{y}{z}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+18}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -6.4 \cdot 10^{-242}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-233}:\\ \;\;\;\;\frac{x\_m \cdot y}{t}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-60}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* x_m (/ (- y z) t))) (t_2 (* x_m (- 1.0 (/ y z)))))
   (*
    x_s
    (if (<= z -5.6e+18)
      t_2
      (if (<= z -6.4e-242)
        t_1
        (if (<= z 6e-233) (/ (* x_m y) t) (if (<= z 8e-60) t_1 t_2)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * ((y - z) / t);
	double t_2 = x_m * (1.0 - (y / z));
	double tmp;
	if (z <= -5.6e+18) {
		tmp = t_2;
	} else if (z <= -6.4e-242) {
		tmp = t_1;
	} else if (z <= 6e-233) {
		tmp = (x_m * y) / t;
	} else if (z <= 8e-60) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x_m * ((y - z) / t)
    t_2 = x_m * (1.0d0 - (y / z))
    if (z <= (-5.6d+18)) then
        tmp = t_2
    else if (z <= (-6.4d-242)) then
        tmp = t_1
    else if (z <= 6d-233) then
        tmp = (x_m * y) / t
    else if (z <= 8d-60) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * ((y - z) / t);
	double t_2 = x_m * (1.0 - (y / z));
	double tmp;
	if (z <= -5.6e+18) {
		tmp = t_2;
	} else if (z <= -6.4e-242) {
		tmp = t_1;
	} else if (z <= 6e-233) {
		tmp = (x_m * y) / t;
	} else if (z <= 8e-60) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = x_m * ((y - z) / t)
	t_2 = x_m * (1.0 - (y / z))
	tmp = 0
	if z <= -5.6e+18:
		tmp = t_2
	elif z <= -6.4e-242:
		tmp = t_1
	elif z <= 6e-233:
		tmp = (x_m * y) / t
	elif z <= 8e-60:
		tmp = t_1
	else:
		tmp = t_2
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m * Float64(Float64(y - z) / t))
	t_2 = Float64(x_m * Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (z <= -5.6e+18)
		tmp = t_2;
	elseif (z <= -6.4e-242)
		tmp = t_1;
	elseif (z <= 6e-233)
		tmp = Float64(Float64(x_m * y) / t);
	elseif (z <= 8e-60)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m * ((y - z) / t);
	t_2 = x_m * (1.0 - (y / z));
	tmp = 0.0;
	if (z <= -5.6e+18)
		tmp = t_2;
	elseif (z <= -6.4e-242)
		tmp = t_1;
	elseif (z <= 6e-233)
		tmp = (x_m * y) / t;
	elseif (z <= 8e-60)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m * N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x$95$m * N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -5.6e+18], t$95$2, If[LessEqual[z, -6.4e-242], t$95$1, If[LessEqual[z, 6e-233], N[(N[(x$95$m * y), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 8e-60], t$95$1, t$95$2]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.6e18 or 7.9999999999999998e-60 < z

   1. Initial program 74.6%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 64.0%

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

      1. mul-1-neg 64.0%

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

      2. associate-/l* 82.3%

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

      3. distribute-rgt-neg-in 82.3%

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

      4. distribute-frac-neg 82.3%

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

      5. neg-sub0 82.3%

         \[\leadsto x \cdot \frac{\color{blue}{0 - \left(y - z\right)}}{z} \]

      6. associate--r- 82.3%

         \[\leadsto x \cdot \frac{\color{blue}{\left(0 - y\right) + z}}{z} \]

      7. neg-sub0 82.3%

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

      8. +-commutative 82.3%

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

      9. sub-neg 82.3%

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

      10. div-sub 82.3%

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

      11. *-inverses 82.3%

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

   7. Simplified 82.3%

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

   if -5.6e18 < z < -6.39999999999999997e-242 or 5.99999999999999997e-233 < z < 7.9999999999999998e-60

   1. Initial program 89.1%

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

      1. associate-/l* 95.4%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in t around inf 77.9%

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

      1. associate-/l* 82.0%

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

   7. Simplified 82.0%

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

   if -6.39999999999999997e-242 < z < 5.99999999999999997e-233

   1. Initial program 97.1%

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

      1. associate-/l* 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in z around 0 83.6%

      \[\leadsto \color{blue}{\frac{x \cdot y}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq -6.4 \cdot 10^{-242}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-233}:\\ \;\;\;\;\frac{x \cdot y}{t}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-60}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \left(1 - \frac{y}{z}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.75 \cdot 10^{-201}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x\_m}{t}\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-209}:\\ \;\;\;\;\frac{x\_m \cdot y}{t - z}\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{x\_m}{\frac{t}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* x_m (- 1.0 (/ y z)))))
   (*
    x_s
    (if (<= z -2.9e+16)
      t_1
      (if (<= z -2.75e-201)
        (* (- y z) (/ x_m t))
        (if (<= z 2.15e-209)
          (/ (* x_m y) (- t z))
          (if (<= z 7.6e-60) (/ x_m (/ t (- y z))) t_1)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * (1.0 - (y / z));
	double tmp;
	if (z <= -2.9e+16) {
		tmp = t_1;
	} else if (z <= -2.75e-201) {
		tmp = (y - z) * (x_m / t);
	} else if (z <= 2.15e-209) {
		tmp = (x_m * y) / (t - z);
	} else if (z <= 7.6e-60) {
		tmp = x_m / (t / (y - z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m * (1.0d0 - (y / z))
    if (z <= (-2.9d+16)) then
        tmp = t_1
    else if (z <= (-2.75d-201)) then
        tmp = (y - z) * (x_m / t)
    else if (z <= 2.15d-209) then
        tmp = (x_m * y) / (t - z)
    else if (z <= 7.6d-60) then
        tmp = x_m / (t / (y - z))
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * (1.0 - (y / z));
	double tmp;
	if (z <= -2.9e+16) {
		tmp = t_1;
	} else if (z <= -2.75e-201) {
		tmp = (y - z) * (x_m / t);
	} else if (z <= 2.15e-209) {
		tmp = (x_m * y) / (t - z);
	} else if (z <= 7.6e-60) {
		tmp = x_m / (t / (y - z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = x_m * (1.0 - (y / z))
	tmp = 0
	if z <= -2.9e+16:
		tmp = t_1
	elif z <= -2.75e-201:
		tmp = (y - z) * (x_m / t)
	elif z <= 2.15e-209:
		tmp = (x_m * y) / (t - z)
	elif z <= 7.6e-60:
		tmp = x_m / (t / (y - z))
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m * Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (z <= -2.9e+16)
		tmp = t_1;
	elseif (z <= -2.75e-201)
		tmp = Float64(Float64(y - z) * Float64(x_m / t));
	elseif (z <= 2.15e-209)
		tmp = Float64(Float64(x_m * y) / Float64(t - z));
	elseif (z <= 7.6e-60)
		tmp = Float64(x_m / Float64(t / Float64(y - z)));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m * (1.0 - (y / z));
	tmp = 0.0;
	if (z <= -2.9e+16)
		tmp = t_1;
	elseif (z <= -2.75e-201)
		tmp = (y - z) * (x_m / t);
	elseif (z <= 2.15e-209)
		tmp = (x_m * y) / (t - z);
	elseif (z <= 7.6e-60)
		tmp = x_m / (t / (y - z));
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m * N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -2.9e+16], t$95$1, If[LessEqual[z, -2.75e-201], N[(N[(y - z), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.15e-209], N[(N[(x$95$m * y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.6e-60], N[(x$95$m / N[(t / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if z < -2.9e16 or 7.59999999999999989e-60 < z

   1. Initial program 74.6%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 64.0%

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

      1. mul-1-neg 64.0%

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

      2. associate-/l* 82.3%

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

      3. distribute-rgt-neg-in 82.3%

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

      4. distribute-frac-neg 82.3%

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

      5. neg-sub0 82.3%

         \[\leadsto x \cdot \frac{\color{blue}{0 - \left(y - z\right)}}{z} \]

      6. associate--r- 82.3%

         \[\leadsto x \cdot \frac{\color{blue}{\left(0 - y\right) + z}}{z} \]

      7. neg-sub0 82.3%

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

      8. +-commutative 82.3%

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

      9. sub-neg 82.3%

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

      10. div-sub 82.3%

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

      11. *-inverses 82.3%

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

   7. Simplified 82.3%

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

   if -2.9e16 < z < -2.75000000000000017e-201

   1. Initial program 88.9%

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

      1. associate-/l* 93.9%

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

   3. Simplified 93.9%

      \[\leadsto \color{blue}{x \cdot \frac{y - z}{t - z}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 91.1%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{t - z}{y - z}}} \]

      2. un-div-inv 91.3%

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

   6. Applied egg-rr 91.3%

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

   7. Taylor expanded in t around inf 76.6%

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

      1. associate-/r/ 79.5%

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

   9. Applied egg-rr 79.5%

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

   if -2.75000000000000017e-201 < z < 2.15000000000000003e-209

   1. Initial program 97.9%

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

      1. associate-/l* 83.9%

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

   3. Simplified 83.9%

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

   5. Taylor expanded in y around inf 94.4%

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

   if 2.15000000000000003e-209 < z < 7.59999999999999989e-60

   1. Initial program 84.3%

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

      1. associate-/l* 94.6%

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

   3. Simplified 94.6%

      \[\leadsto \color{blue}{x \cdot \frac{y - z}{t - z}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 94.4%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{t - z}{y - z}}} \]

      2. un-div-inv 94.6%

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

   6. Applied egg-rr 94.6%

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

   7. Taylor expanded in t around inf 83.5%

      \[\leadsto \frac{x}{\color{blue}{\frac{t}{y - z}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+16}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq -2.75 \cdot 10^{-201}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-209}:\\ \;\;\;\;\frac{x \cdot y}{t - z}\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{\frac{t}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+17} \lor \neg \left(z \leq 7.5 \cdot 10^{-61}\right):\\ \;\;\;\;x\_m \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -1.8e+17) (not (<= z 7.5e-61)))
    (* x_m (- 1.0 (/ y z)))
    (* y (/ x_m t)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -1.8e+17) || !(z <= 7.5e-61)) {
		tmp = x_m * (1.0 - (y / z));
	} else {
		tmp = y * (x_m / t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.8d+17)) .or. (.not. (z <= 7.5d-61))) then
        tmp = x_m * (1.0d0 - (y / z))
    else
        tmp = y * (x_m / t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -1.8e+17) || !(z <= 7.5e-61)) {
		tmp = x_m * (1.0 - (y / z));
	} else {
		tmp = y * (x_m / t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (z <= -1.8e+17) or not (z <= 7.5e-61):
		tmp = x_m * (1.0 - (y / z))
	else:
		tmp = y * (x_m / t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((z <= -1.8e+17) || !(z <= 7.5e-61))
		tmp = Float64(x_m * Float64(1.0 - Float64(y / z)));
	else
		tmp = Float64(y * Float64(x_m / t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((z <= -1.8e+17) || ~((z <= 7.5e-61)))
		tmp = x_m * (1.0 - (y / z));
	else
		tmp = y * (x_m / t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -1.8e+17], N[Not[LessEqual[z, 7.5e-61]], $MachinePrecision]], N[(x$95$m * N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.8e17 or 7.50000000000000047e-61 < z

   1. Initial program 74.6%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 64.0%

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

      1. mul-1-neg 64.0%

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

      2. associate-/l* 82.3%

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

      3. distribute-rgt-neg-in 82.3%

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

      4. distribute-frac-neg 82.3%

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

      5. neg-sub0 82.3%

         \[\leadsto x \cdot \frac{\color{blue}{0 - \left(y - z\right)}}{z} \]

      6. associate--r- 82.3%

         \[\leadsto x \cdot \frac{\color{blue}{\left(0 - y\right) + z}}{z} \]

      7. neg-sub0 82.3%

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

      8. +-commutative 82.3%

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

      9. sub-neg 82.3%

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

      10. div-sub 82.3%

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

      11. *-inverses 82.3%

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

   7. Simplified 82.3%

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

   if -1.8e17 < z < 7.50000000000000047e-61

   1. Initial program 91.3%

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

      1. associate-/l* 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in y around inf 77.1%

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

      1. *-commutative 77.1%

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

      2. associate-/l* 80.2%

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

   7. Applied egg-rr 80.2%

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

   8. Taylor expanded in t around inf 71.6%

      \[\leadsto y \cdot \color{blue}{\frac{x}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+17} \lor \neg \left(z \leq 7.5 \cdot 10^{-61}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+17} \lor \neg \left(z \leq 4.6 \cdot 10^{-29}\right):\\ \;\;\;\;x\_m \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{t - z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -2.6e+17) (not (<= z 4.6e-29)))
    (* x_m (- 1.0 (/ y z)))
    (* y (/ x_m (- t z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -2.6e+17) || !(z <= 4.6e-29)) {
		tmp = x_m * (1.0 - (y / z));
	} else {
		tmp = y * (x_m / (t - z));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-2.6d+17)) .or. (.not. (z <= 4.6d-29))) then
        tmp = x_m * (1.0d0 - (y / z))
    else
        tmp = y * (x_m / (t - z))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -2.6e+17) || !(z <= 4.6e-29)) {
		tmp = x_m * (1.0 - (y / z));
	} else {
		tmp = y * (x_m / (t - z));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (z <= -2.6e+17) or not (z <= 4.6e-29):
		tmp = x_m * (1.0 - (y / z))
	else:
		tmp = y * (x_m / (t - z))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((z <= -2.6e+17) || !(z <= 4.6e-29))
		tmp = Float64(x_m * Float64(1.0 - Float64(y / z)));
	else
		tmp = Float64(y * Float64(x_m / Float64(t - z)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((z <= -2.6e+17) || ~((z <= 4.6e-29)))
		tmp = x_m * (1.0 - (y / z));
	else
		tmp = y * (x_m / (t - z));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -2.6e+17], N[Not[LessEqual[z, 4.6e-29]], $MachinePrecision]], N[(x$95$m * N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -2.6e17 or 4.59999999999999982e-29 < z

   1. Initial program 74.1%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 63.9%

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

      1. mul-1-neg 63.9%

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

      2. associate-/l* 82.6%

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

      3. distribute-rgt-neg-in 82.6%

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

      4. distribute-frac-neg 82.6%

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

      5. neg-sub0 82.6%

         \[\leadsto x \cdot \frac{\color{blue}{0 - \left(y - z\right)}}{z} \]

      6. associate--r- 82.6%

         \[\leadsto x \cdot \frac{\color{blue}{\left(0 - y\right) + z}}{z} \]

      7. neg-sub0 82.6%

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

      8. +-commutative 82.6%

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

      9. sub-neg 82.6%

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

      10. div-sub 82.6%

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

      11. *-inverses 82.6%

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

   7. Simplified 82.6%

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

   if -2.6e17 < z < 4.59999999999999982e-29

   1. Initial program 91.5%

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

      1. associate-/l* 90.1%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in y around inf 77.7%

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

      1. *-commutative 77.7%

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

      2. associate-/l* 80.7%

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

   7. Applied egg-rr 80.7%

      \[\leadsto \color{blue}{y \cdot \frac{x}{t - z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+17} \lor \neg \left(z \leq 4.6 \cdot 10^{-29}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+15} \lor \neg \left(z \leq 7.6 \cdot 10^{-60}\right):\\ \;\;\;\;x\_m \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x\_m}{t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -3.6e+15) (not (<= z 7.6e-60)))
    (* x_m (- 1.0 (/ y z)))
    (* (- y z) (/ x_m t)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -3.6e+15) || !(z <= 7.6e-60)) {
		tmp = x_m * (1.0 - (y / z));
	} else {
		tmp = (y - z) * (x_m / t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-3.6d+15)) .or. (.not. (z <= 7.6d-60))) then
        tmp = x_m * (1.0d0 - (y / z))
    else
        tmp = (y - z) * (x_m / t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -3.6e+15) || !(z <= 7.6e-60)) {
		tmp = x_m * (1.0 - (y / z));
	} else {
		tmp = (y - z) * (x_m / t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (z <= -3.6e+15) or not (z <= 7.6e-60):
		tmp = x_m * (1.0 - (y / z))
	else:
		tmp = (y - z) * (x_m / t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((z <= -3.6e+15) || !(z <= 7.6e-60))
		tmp = Float64(x_m * Float64(1.0 - Float64(y / z)));
	else
		tmp = Float64(Float64(y - z) * Float64(x_m / t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((z <= -3.6e+15) || ~((z <= 7.6e-60)))
		tmp = x_m * (1.0 - (y / z));
	else
		tmp = (y - z) * (x_m / t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -3.6e+15], N[Not[LessEqual[z, 7.6e-60]], $MachinePrecision]], N[(x$95$m * N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.6e15 or 7.59999999999999989e-60 < z

   1. Initial program 74.6%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 64.0%

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

      1. mul-1-neg 64.0%

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

      2. associate-/l* 82.3%

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

      3. distribute-rgt-neg-in 82.3%

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

      4. distribute-frac-neg 82.3%

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

      5. neg-sub0 82.3%

         \[\leadsto x \cdot \frac{\color{blue}{0 - \left(y - z\right)}}{z} \]

      6. associate--r- 82.3%

         \[\leadsto x \cdot \frac{\color{blue}{\left(0 - y\right) + z}}{z} \]

      7. neg-sub0 82.3%

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

      8. +-commutative 82.3%

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

      9. sub-neg 82.3%

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

      10. div-sub 82.3%

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

      11. *-inverses 82.3%

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

   7. Simplified 82.3%

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

   if -3.6e15 < z < 7.59999999999999989e-60

   1. Initial program 91.3%

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

      1. associate-/l* 89.9%

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

   3. Simplified 89.9%

      \[\leadsto \color{blue}{x \cdot \frac{y - z}{t - z}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 88.2%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{t - z}{y - z}}} \]

      2. un-div-inv 89.0%

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

   6. Applied egg-rr 89.0%

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

   7. Taylor expanded in t around inf 77.4%

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

      1. associate-/r/ 81.6%

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

   9. Applied egg-rr 81.6%

      \[\leadsto \color{blue}{\frac{x}{t} \cdot \left(y - z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+15} \lor \neg \left(z \leq 7.6 \cdot 10^{-60}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+17}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+18}:\\ \;\;\;\;x\_m \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (if (<= z -5.6e+17) x_m (if (<= z 5.4e+18) (* x_m (/ y t)) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -5.6e+17) {
		tmp = x_m;
	} else if (z <= 5.4e+18) {
		tmp = x_m * (y / t);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-5.6d+17)) then
        tmp = x_m
    else if (z <= 5.4d+18) then
        tmp = x_m * (y / t)
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -5.6e+17) {
		tmp = x_m;
	} else if (z <= 5.4e+18) {
		tmp = x_m * (y / t);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -5.6e+17:
		tmp = x_m
	elif z <= 5.4e+18:
		tmp = x_m * (y / t)
	else:
		tmp = x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -5.6e+17)
		tmp = x_m;
	elseif (z <= 5.4e+18)
		tmp = Float64(x_m * Float64(y / t));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -5.6e+17)
		tmp = x_m;
	elseif (z <= 5.4e+18)
		tmp = x_m * (y / t);
	else
		tmp = x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -5.6e+17], x$95$m, If[LessEqual[z, 5.4e+18], N[(x$95$m * N[(y / t), $MachinePrecision]), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -5.6e17 or 5.4e18 < z

   1. Initial program 72.3%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 60.4%

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

   if -5.6e17 < z < 5.4e18

   1. Initial program 91.5%

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

      1. associate-/l* 91.0%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in z around 0 65.1%

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

      1. associate-/l* 67.0%

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

   7. Simplified 67.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+17}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+16}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+18}:\\ \;\;\;\;y \cdot \frac{x\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (if (<= z -1.9e+16) x_m (if (<= z 3.4e+18) (* y (/ x_m t)) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -1.9e+16) {
		tmp = x_m;
	} else if (z <= 3.4e+18) {
		tmp = y * (x_m / t);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.9d+16)) then
        tmp = x_m
    else if (z <= 3.4d+18) then
        tmp = y * (x_m / t)
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -1.9e+16) {
		tmp = x_m;
	} else if (z <= 3.4e+18) {
		tmp = y * (x_m / t);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -1.9e+16:
		tmp = x_m
	elif z <= 3.4e+18:
		tmp = y * (x_m / t)
	else:
		tmp = x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -1.9e+16)
		tmp = x_m;
	elseif (z <= 3.4e+18)
		tmp = Float64(y * Float64(x_m / t));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -1.9e+16)
		tmp = x_m;
	elseif (z <= 3.4e+18)
		tmp = y * (x_m / t);
	else
		tmp = x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -1.9e+16], x$95$m, If[LessEqual[z, 3.4e+18], N[(y * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.9e16 or 3.4e18 < z

   1. Initial program 72.3%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 60.4%

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

   if -1.9e16 < z < 3.4e18

   1. Initial program 91.5%

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

      1. associate-/l* 91.0%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in y around inf 75.4%

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

      1. *-commutative 75.4%

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

      2. associate-/l* 78.2%

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

   7. Applied egg-rr 78.2%

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

   8. Taylor expanded in t around inf 67.7%

      \[\leadsto y \cdot \color{blue}{\frac{x}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+18}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4.4 \cdot 10^{-26}:\\ \;\;\;\;x\_m \cdot \frac{y - z}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z - t} \cdot \left(z - y\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= x_m 4.4e-26)
    (* x_m (/ (- y z) (- t z)))
    (* (/ x_m (- z t)) (- z y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 4.4e-26) {
		tmp = x_m * ((y - z) / (t - z));
	} else {
		tmp = (x_m / (z - t)) * (z - y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x_m <= 4.4d-26) then
        tmp = x_m * ((y - z) / (t - z))
    else
        tmp = (x_m / (z - t)) * (z - y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 4.4e-26) {
		tmp = x_m * ((y - z) / (t - z));
	} else {
		tmp = (x_m / (z - t)) * (z - y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if x_m <= 4.4e-26:
		tmp = x_m * ((y - z) / (t - z))
	else:
		tmp = (x_m / (z - t)) * (z - y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 4.4e-26)
		tmp = Float64(x_m * Float64(Float64(y - z) / Float64(t - z)));
	else
		tmp = Float64(Float64(x_m / Float64(z - t)) * Float64(z - y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (x_m <= 4.4e-26)
		tmp = x_m * ((y - z) / (t - z));
	else
		tmp = (x_m / (z - t)) * (z - y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[x$95$m, 4.4e-26], N[(x$95$m * N[(N[(y - z), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(z - t), $MachinePrecision]), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.4000000000000002e-26

   1. Initial program 88.1%

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

      1. associate-/l* 95.1%

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

   3. Simplified 95.1%

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

   if 4.4000000000000002e-26 < x

   1. Initial program 65.4%

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

      1. associate-/l* 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in x around 0 65.4%

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

      1. remove-double-neg 65.4%

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

      2. distribute-neg-frac2 65.4%

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

      3. *-commutative 65.4%

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

      4. associate-/l* 98.2%

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

      5. distribute-lft-neg-out 98.2%

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

      6. neg-sub0 98.2%

         \[\leadsto \color{blue}{\left(0 - \left(y - z\right)\right)} \cdot \frac{x}{-\left(t - z\right)} \]

      7. associate--r- 98.2%

         \[\leadsto \color{blue}{\left(\left(0 - y\right) + z\right)} \cdot \frac{x}{-\left(t - z\right)} \]

      8. neg-sub0 98.2%

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

      9. +-commutative 98.2%

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

      10. sub-neg 98.2%

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

      11. neg-sub0 98.2%

          \[\leadsto \left(z - y\right) \cdot \frac{x}{\color{blue}{0 - \left(t - z\right)}} \]

      12. associate--r- 98.2%

          \[\leadsto \left(z - y\right) \cdot \frac{x}{\color{blue}{\left(0 - t\right) + z}} \]

      13. neg-sub0 98.2%

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

      14. +-commutative 98.2%

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

      15. sub-neg 98.2%

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

      16. *-commutative 98.2%

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

   7. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{x}{z - t} \cdot \left(z - y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{-26}:\\ \;\;\;\;x \cdot \frac{y - z}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - t} \cdot \left(z - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{x\_m}{\frac{t - z}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z - t} \cdot \left(z - y\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= x_m 3.5e-26)
    (/ x_m (/ (- t z) (- y z)))
    (* (/ x_m (- z t)) (- z y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 3.5e-26) {
		tmp = x_m / ((t - z) / (y - z));
	} else {
		tmp = (x_m / (z - t)) * (z - y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x_m <= 3.5d-26) then
        tmp = x_m / ((t - z) / (y - z))
    else
        tmp = (x_m / (z - t)) * (z - y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 3.5e-26) {
		tmp = x_m / ((t - z) / (y - z));
	} else {
		tmp = (x_m / (z - t)) * (z - y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if x_m <= 3.5e-26:
		tmp = x_m / ((t - z) / (y - z))
	else:
		tmp = (x_m / (z - t)) * (z - y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 3.5e-26)
		tmp = Float64(x_m / Float64(Float64(t - z) / Float64(y - z)));
	else
		tmp = Float64(Float64(x_m / Float64(z - t)) * Float64(z - y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (x_m <= 3.5e-26)
		tmp = x_m / ((t - z) / (y - z));
	else
		tmp = (x_m / (z - t)) * (z - y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[x$95$m, 3.5e-26], N[(x$95$m / N[(N[(t - z), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(z - t), $MachinePrecision]), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.49999999999999985e-26

   1. Initial program 88.1%

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

      1. associate-/l* 95.1%

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

   3. Simplified 95.1%

      \[\leadsto \color{blue}{x \cdot \frac{y - z}{t - z}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 94.3%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{t - z}{y - z}}} \]

      2. un-div-inv 94.7%

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

   6. Applied egg-rr 94.7%

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

   if 3.49999999999999985e-26 < x

   1. Initial program 65.4%

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

      1. associate-/l* 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in x around 0 65.4%

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

      1. remove-double-neg 65.4%

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

      2. distribute-neg-frac2 65.4%

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

      3. *-commutative 65.4%

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

      4. associate-/l* 98.2%

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

      5. distribute-lft-neg-out 98.2%

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

      6. neg-sub0 98.2%

         \[\leadsto \color{blue}{\left(0 - \left(y - z\right)\right)} \cdot \frac{x}{-\left(t - z\right)} \]

      7. associate--r- 98.2%

         \[\leadsto \color{blue}{\left(\left(0 - y\right) + z\right)} \cdot \frac{x}{-\left(t - z\right)} \]

      8. neg-sub0 98.2%

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

      9. +-commutative 98.2%

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

      10. sub-neg 98.2%

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

      11. neg-sub0 98.2%

          \[\leadsto \left(z - y\right) \cdot \frac{x}{\color{blue}{0 - \left(t - z\right)}} \]

      12. associate--r- 98.2%

          \[\leadsto \left(z - y\right) \cdot \frac{x}{\color{blue}{\left(0 - t\right) + z}} \]

      13. neg-sub0 98.2%

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

      14. +-commutative 98.2%

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

      15. sub-neg 98.2%

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

      16. *-commutative 98.2%

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

   7. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{x}{z - t} \cdot \left(z - y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - t} \cdot \left(z - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.9% accurate, 1.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (* x_m (/ (- y z) (- t z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m * ((y - z) / (t - z)));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x_s * (x_m * ((y - z) / (t - z)))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m * ((y - z) / (t - z)));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	return x_s * (x_m * ((y - z) / (t - z)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(x_m * Float64(Float64(y - z) / Float64(t - z))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * (x_m * ((y - z) / (t - z)));
end

Wolfram

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

Derivation

1. Initial program 82.5%

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

   1. associate-/l* 95.2%

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

3. Simplified 95.2%

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

5. Final simplification 95.2%

   \[\leadsto x \cdot \frac{y - z}{t - z} \]
6. Add Preprocessing

Alternative 12: 34.8% accurate, 9.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t) :precision binary64 (* x_s x_m))

C

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

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x_s * x_m
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

   1. associate-/l* 95.2%

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

3. Simplified 95.2%

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

5. Taylor expanded in z around inf 33.3%

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

6. Final simplification 33.3%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{t - z}{y - z}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x / ((t - z) / (y - z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x / ((t - z) / (y - z));
}

Python

def code(x, y, z, t):
	return x / ((t - z) / (y - z))

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024096 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

  :alt
  (/ x (/ (- t z) (- y z)))

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