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

Percentage Accurate: 83.7% → 97.0%

Time: 11.0s

Alternatives: 14

Speedup: 0.8×

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: 83.7% 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: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{x\_m \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x\_m}{t - z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5e-5) (/ (* x_m (- y z)) (- t z)) (* (- y z) (/ 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 (x_m <= 5e-5) {
		tmp = (x_m * (y - z)) / (t - z);
	} else {
		tmp = (y - z) * (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 (x_m <= 5d-5) then
        tmp = (x_m * (y - z)) / (t - z)
    else
        tmp = (y - z) * (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 (x_m <= 5e-5) {
		tmp = (x_m * (y - z)) / (t - z);
	} else {
		tmp = (y - z) * (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 x_m <= 5e-5:
		tmp = (x_m * (y - z)) / (t - z)
	else:
		tmp = (y - z) * (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 (x_m <= 5e-5)
		tmp = Float64(Float64(x_m * Float64(y - z)) / Float64(t - z));
	else
		tmp = Float64(Float64(y - z) * 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 (x_m <= 5e-5)
		tmp = (x_m * (y - z)) / (t - z);
	else
		tmp = (y - z) * (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[LessEqual[x$95$m, 5e-5], N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000024e-5

   1. Initial program 86.8%

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

   if 5.00000000000000024e-5 < x

   1. Initial program 67.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 90.3%

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

Alternative 2: 90.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \left(y - z\right) \cdot \frac{x\_m}{t - z}\\ t_2 := x\_m \cdot \left(y - z\right)\\ t_3 := \frac{t\_2}{t - z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -1 \cdot 10^{-141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 10^{-310}:\\ \;\;\;\;\frac{t\_2}{t}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-98}:\\ \;\;\;\;\frac{x\_m \cdot z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* (- y z) (/ x_m (- t z))))
        (t_2 (* x_m (- y z)))
        (t_3 (/ t_2 (- t z))))
   (*
    x_s
    (if (<= t_3 -1e-141)
      t_1
      (if (<= t_3 1e-310)
        (/ t_2 t)
        (if (<= t_3 2e-98) (/ (* x_m z) (- z t)) 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 = (y - z) * (x_m / (t - z));
	double t_2 = x_m * (y - z);
	double t_3 = t_2 / (t - z);
	double tmp;
	if (t_3 <= -1e-141) {
		tmp = t_1;
	} else if (t_3 <= 1e-310) {
		tmp = t_2 / t;
	} else if (t_3 <= 2e-98) {
		tmp = (x_m * z) / (z - t);
	} 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (y - z) * (x_m / (t - z))
    t_2 = x_m * (y - z)
    t_3 = t_2 / (t - z)
    if (t_3 <= (-1d-141)) then
        tmp = t_1
    else if (t_3 <= 1d-310) then
        tmp = t_2 / t
    else if (t_3 <= 2d-98) then
        tmp = (x_m * z) / (z - t)
    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 = (y - z) * (x_m / (t - z));
	double t_2 = x_m * (y - z);
	double t_3 = t_2 / (t - z);
	double tmp;
	if (t_3 <= -1e-141) {
		tmp = t_1;
	} else if (t_3 <= 1e-310) {
		tmp = t_2 / t;
	} else if (t_3 <= 2e-98) {
		tmp = (x_m * z) / (z - t);
	} 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 = (y - z) * (x_m / (t - z))
	t_2 = x_m * (y - z)
	t_3 = t_2 / (t - z)
	tmp = 0
	if t_3 <= -1e-141:
		tmp = t_1
	elif t_3 <= 1e-310:
		tmp = t_2 / t
	elif t_3 <= 2e-98:
		tmp = (x_m * z) / (z - t)
	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(Float64(y - z) * Float64(x_m / Float64(t - z)))
	t_2 = Float64(x_m * Float64(y - z))
	t_3 = Float64(t_2 / Float64(t - z))
	tmp = 0.0
	if (t_3 <= -1e-141)
		tmp = t_1;
	elseif (t_3 <= 1e-310)
		tmp = Float64(t_2 / t);
	elseif (t_3 <= 2e-98)
		tmp = Float64(Float64(x_m * z) / Float64(z - t));
	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 = (y - z) * (x_m / (t - z));
	t_2 = x_m * (y - z);
	t_3 = t_2 / (t - z);
	tmp = 0.0;
	if (t_3 <= -1e-141)
		tmp = t_1;
	elseif (t_3 <= 1e-310)
		tmp = t_2 / t;
	elseif (t_3 <= 2e-98)
		tmp = (x_m * z) / (z - t);
	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[(N[(y - z), $MachinePrecision] * N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(t - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$3, -1e-141], t$95$1, If[LessEqual[t$95$3, 1e-310], N[(t$95$2 / t), $MachinePrecision], If[LessEqual[t$95$3, 2e-98], N[(N[(x$95$m * z), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -1e-141 or 1.99999999999999988e-98 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 73.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 96.5

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

   4. Applied egg-rr 96.5%

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

   if -1e-141 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 9.999999999999969e-311

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 76.9

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

   5. Simplified 76.9%

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

   if 9.999999999999969e-311 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 1.99999999999999988e-98

   1. Initial program 99.7%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{t - z}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. mul-1-neg N/A

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

      10. /-lowering-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. --lowering--.f64 33.8

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

   7. Simplified 33.8%

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

      1. associate-*r/ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. --lowering--.f64 65.8

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

   9. Applied egg-rr 65.8%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -1 \cdot 10^{-141}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t - z}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 10^{-310}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 2 \cdot 10^{-98}:\\ \;\;\;\;\frac{x \cdot z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \left(y - z\right)\\ t_2 := \frac{t\_1}{t - z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-114}:\\ \;\;\;\;y \cdot \frac{x\_m}{t - z}\\ \mathbf{elif}\;t\_2 \leq 10^{-310}:\\ \;\;\;\;\frac{t\_1}{t}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-75}:\\ \;\;\;\;\frac{x\_m \cdot z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;x\_m - y \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* x_m (- y z))) (t_2 (/ t_1 (- t z))))
   (*
    x_s
    (if (<= t_2 -2e-114)
      (* y (/ x_m (- t z)))
      (if (<= t_2 1e-310)
        (/ t_1 t)
        (if (<= t_2 2e-75) (/ (* x_m z) (- z t)) (- x_m (* y (/ x_m 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 t_1 = x_m * (y - z);
	double t_2 = t_1 / (t - z);
	double tmp;
	if (t_2 <= -2e-114) {
		tmp = y * (x_m / (t - z));
	} else if (t_2 <= 1e-310) {
		tmp = t_1 / t;
	} else if (t_2 <= 2e-75) {
		tmp = (x_m * z) / (z - t);
	} else {
		tmp = x_m - (y * (x_m / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x_m * (y - z)
    t_2 = t_1 / (t - z)
    if (t_2 <= (-2d-114)) then
        tmp = y * (x_m / (t - z))
    else if (t_2 <= 1d-310) then
        tmp = t_1 / t
    else if (t_2 <= 2d-75) then
        tmp = (x_m * z) / (z - t)
    else
        tmp = x_m - (y * (x_m / 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 t_1 = x_m * (y - z);
	double t_2 = t_1 / (t - z);
	double tmp;
	if (t_2 <= -2e-114) {
		tmp = y * (x_m / (t - z));
	} else if (t_2 <= 1e-310) {
		tmp = t_1 / t;
	} else if (t_2 <= 2e-75) {
		tmp = (x_m * z) / (z - t);
	} else {
		tmp = x_m - (y * (x_m / 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):
	t_1 = x_m * (y - z)
	t_2 = t_1 / (t - z)
	tmp = 0
	if t_2 <= -2e-114:
		tmp = y * (x_m / (t - z))
	elif t_2 <= 1e-310:
		tmp = t_1 / t
	elif t_2 <= 2e-75:
		tmp = (x_m * z) / (z - t)
	else:
		tmp = x_m - (y * (x_m / 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)
	t_1 = Float64(x_m * Float64(y - z))
	t_2 = Float64(t_1 / Float64(t - z))
	tmp = 0.0
	if (t_2 <= -2e-114)
		tmp = Float64(y * Float64(x_m / Float64(t - z)));
	elseif (t_2 <= 1e-310)
		tmp = Float64(t_1 / t);
	elseif (t_2 <= 2e-75)
		tmp = Float64(Float64(x_m * z) / Float64(z - t));
	else
		tmp = Float64(x_m - Float64(y * Float64(x_m / 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)
	t_1 = x_m * (y - z);
	t_2 = t_1 / (t - z);
	tmp = 0.0;
	if (t_2 <= -2e-114)
		tmp = y * (x_m / (t - z));
	elseif (t_2 <= 1e-310)
		tmp = t_1 / t;
	elseif (t_2 <= 2e-75)
		tmp = (x_m * z) / (z - t);
	else
		tmp = x_m - (y * (x_m / 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_] := Block[{t$95$1 = N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(t - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$2, -2e-114], N[(y * N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-310], N[(t$95$1 / t), $MachinePrecision], If[LessEqual[t$95$2, 2e-75], N[(N[(x$95$m * z), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], N[(x$95$m - N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -2.0000000000000001e-114

   1. Initial program 73.5%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 46.2

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

   7. Simplified 46.2%

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

      1. associate-/r/ N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 45.1

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

   9. Applied egg-rr 45.1%

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

   if -2.0000000000000001e-114 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 9.999999999999969e-311

   1. Initial program 99.8%

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

   3. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 74.7

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

   5. Simplified 74.7%

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

   if 9.999999999999969e-311 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 1.9999999999999999e-75

   1. Initial program 99.6%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{t - z}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. mul-1-neg N/A

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

      10. /-lowering-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. --lowering--.f64 34.9

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

   7. Simplified 34.9%

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

      1. associate-*r/ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. --lowering--.f64 65.0

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

   9. Applied egg-rr 65.0%

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

   if 1.9999999999999999e-75 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 72.3%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. associate-/l* N/A

         \[\leadsto 0 - \color{blue}{x \cdot \frac{y - z}{z}} \]

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \color{blue}{-1}\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto 0 - \color{blue}{\left(\frac{y}{z} \cdot x + -1 \cdot x\right)} \]

      9. *-commutative N/A

         \[\leadsto 0 - \left(\color{blue}{x \cdot \frac{y}{z}} + -1 \cdot x\right) \]

      10. associate-/l* N/A

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot y}{z}} + -1 \cdot x\right) \]

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. --lowering--.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/l* N/A

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

      22. *-lowering-*.f64 N/A

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

      23. /-lowering-/.f64 63.7

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

   5. Simplified 63.7%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -2 \cdot 10^{-114}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 10^{-310}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 2 \cdot 10^{-75}:\\ \;\;\;\;\frac{x \cdot z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot \left(y - z\right)}{t - z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-114}:\\ \;\;\;\;y \cdot \frac{x\_m}{t - z}\\ \mathbf{elif}\;t\_1 \leq 10^{-310}:\\ \;\;\;\;x\_m \cdot \frac{y - z}{t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-75}:\\ \;\;\;\;\frac{x\_m \cdot z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;x\_m - y \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (* x_m (- y z)) (- t z))))
   (*
    x_s
    (if (<= t_1 -2e-114)
      (* y (/ x_m (- t z)))
      (if (<= t_1 1e-310)
        (* x_m (/ (- y z) t))
        (if (<= t_1 2e-75) (/ (* x_m z) (- z t)) (- x_m (* y (/ x_m 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 t_1 = (x_m * (y - z)) / (t - z);
	double tmp;
	if (t_1 <= -2e-114) {
		tmp = y * (x_m / (t - z));
	} else if (t_1 <= 1e-310) {
		tmp = x_m * ((y - z) / t);
	} else if (t_1 <= 2e-75) {
		tmp = (x_m * z) / (z - t);
	} else {
		tmp = x_m - (y * (x_m / 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) :: t_1
    real(8) :: tmp
    t_1 = (x_m * (y - z)) / (t - z)
    if (t_1 <= (-2d-114)) then
        tmp = y * (x_m / (t - z))
    else if (t_1 <= 1d-310) then
        tmp = x_m * ((y - z) / t)
    else if (t_1 <= 2d-75) then
        tmp = (x_m * z) / (z - t)
    else
        tmp = x_m - (y * (x_m / 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 t_1 = (x_m * (y - z)) / (t - z);
	double tmp;
	if (t_1 <= -2e-114) {
		tmp = y * (x_m / (t - z));
	} else if (t_1 <= 1e-310) {
		tmp = x_m * ((y - z) / t);
	} else if (t_1 <= 2e-75) {
		tmp = (x_m * z) / (z - t);
	} else {
		tmp = x_m - (y * (x_m / 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):
	t_1 = (x_m * (y - z)) / (t - z)
	tmp = 0
	if t_1 <= -2e-114:
		tmp = y * (x_m / (t - z))
	elif t_1 <= 1e-310:
		tmp = x_m * ((y - z) / t)
	elif t_1 <= 2e-75:
		tmp = (x_m * z) / (z - t)
	else:
		tmp = x_m - (y * (x_m / 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)
	t_1 = Float64(Float64(x_m * Float64(y - z)) / Float64(t - z))
	tmp = 0.0
	if (t_1 <= -2e-114)
		tmp = Float64(y * Float64(x_m / Float64(t - z)));
	elseif (t_1 <= 1e-310)
		tmp = Float64(x_m * Float64(Float64(y - z) / t));
	elseif (t_1 <= 2e-75)
		tmp = Float64(Float64(x_m * z) / Float64(z - t));
	else
		tmp = Float64(x_m - Float64(y * Float64(x_m / 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)
	t_1 = (x_m * (y - z)) / (t - z);
	tmp = 0.0;
	if (t_1 <= -2e-114)
		tmp = y * (x_m / (t - z));
	elseif (t_1 <= 1e-310)
		tmp = x_m * ((y - z) / t);
	elseif (t_1 <= 2e-75)
		tmp = (x_m * z) / (z - t);
	else
		tmp = x_m - (y * (x_m / 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_] := Block[{t$95$1 = N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, -2e-114], N[(y * N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-310], N[(x$95$m * N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-75], N[(N[(x$95$m * z), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], N[(x$95$m - N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -2.0000000000000001e-114

   1. Initial program 73.5%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 46.2

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

   7. Simplified 46.2%

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

      1. associate-/r/ N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 45.1

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

   9. Applied egg-rr 45.1%

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

   if -2.0000000000000001e-114 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 9.999999999999969e-311

   1. Initial program 99.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 66.7

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

   4. Applied egg-rr 66.7%

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

   5. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 62.2

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

   7. Simplified 62.2%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 70.7

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

   9. Applied egg-rr 70.7%

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

   if 9.999999999999969e-311 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 1.9999999999999999e-75

   1. Initial program 99.6%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{t - z}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. mul-1-neg N/A

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

      10. /-lowering-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. --lowering--.f64 34.9

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

   7. Simplified 34.9%

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

      1. associate-*r/ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. --lowering--.f64 65.0

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

   9. Applied egg-rr 65.0%

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

   if 1.9999999999999999e-75 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 72.3%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. associate-/l* N/A

         \[\leadsto 0 - \color{blue}{x \cdot \frac{y - z}{z}} \]

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \color{blue}{-1}\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto 0 - \color{blue}{\left(\frac{y}{z} \cdot x + -1 \cdot x\right)} \]

      9. *-commutative N/A

         \[\leadsto 0 - \left(\color{blue}{x \cdot \frac{y}{z}} + -1 \cdot x\right) \]

      10. associate-/l* N/A

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot y}{z}} + -1 \cdot x\right) \]

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. --lowering--.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/l* N/A

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

      22. *-lowering-*.f64 N/A

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

      23. /-lowering-/.f64 63.7

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

   5. Simplified 63.7%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -2 \cdot 10^{-114}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 10^{-310}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 2 \cdot 10^{-75}:\\ \;\;\;\;\frac{x \cdot z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot \left(y - z\right)}{t - z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-114}:\\ \;\;\;\;y \cdot \frac{x\_m}{t - z}\\ \mathbf{elif}\;t\_1 \leq 10^{-310}:\\ \;\;\;\;x\_m \cdot \frac{y - z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\_m - y \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (* x_m (- y z)) (- t z))))
   (*
    x_s
    (if (<= t_1 -2e-114)
      (* y (/ x_m (- t z)))
      (if (<= t_1 1e-310) (* x_m (/ (- y z) t)) (- x_m (* y (/ x_m 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 t_1 = (x_m * (y - z)) / (t - z);
	double tmp;
	if (t_1 <= -2e-114) {
		tmp = y * (x_m / (t - z));
	} else if (t_1 <= 1e-310) {
		tmp = x_m * ((y - z) / t);
	} else {
		tmp = x_m - (y * (x_m / 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) :: t_1
    real(8) :: tmp
    t_1 = (x_m * (y - z)) / (t - z)
    if (t_1 <= (-2d-114)) then
        tmp = y * (x_m / (t - z))
    else if (t_1 <= 1d-310) then
        tmp = x_m * ((y - z) / t)
    else
        tmp = x_m - (y * (x_m / 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 t_1 = (x_m * (y - z)) / (t - z);
	double tmp;
	if (t_1 <= -2e-114) {
		tmp = y * (x_m / (t - z));
	} else if (t_1 <= 1e-310) {
		tmp = x_m * ((y - z) / t);
	} else {
		tmp = x_m - (y * (x_m / 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):
	t_1 = (x_m * (y - z)) / (t - z)
	tmp = 0
	if t_1 <= -2e-114:
		tmp = y * (x_m / (t - z))
	elif t_1 <= 1e-310:
		tmp = x_m * ((y - z) / t)
	else:
		tmp = x_m - (y * (x_m / 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)
	t_1 = Float64(Float64(x_m * Float64(y - z)) / Float64(t - z))
	tmp = 0.0
	if (t_1 <= -2e-114)
		tmp = Float64(y * Float64(x_m / Float64(t - z)));
	elseif (t_1 <= 1e-310)
		tmp = Float64(x_m * Float64(Float64(y - z) / t));
	else
		tmp = Float64(x_m - Float64(y * Float64(x_m / 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)
	t_1 = (x_m * (y - z)) / (t - z);
	tmp = 0.0;
	if (t_1 <= -2e-114)
		tmp = y * (x_m / (t - z));
	elseif (t_1 <= 1e-310)
		tmp = x_m * ((y - z) / t);
	else
		tmp = x_m - (y * (x_m / 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_] := Block[{t$95$1 = N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, -2e-114], N[(y * N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-310], N[(x$95$m * N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x$95$m - N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -2.0000000000000001e-114

   1. Initial program 73.5%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 46.2

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

   7. Simplified 46.2%

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

      1. associate-/r/ N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 45.1

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

   9. Applied egg-rr 45.1%

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

   if -2.0000000000000001e-114 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 9.999999999999969e-311

   1. Initial program 99.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 66.7

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

   4. Applied egg-rr 66.7%

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

   5. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 62.2

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

   7. Simplified 62.2%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 70.7

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

   9. Applied egg-rr 70.7%

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

   if 9.999999999999969e-311 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 80.1%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. associate-/l* N/A

         \[\leadsto 0 - \color{blue}{x \cdot \frac{y - z}{z}} \]

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \color{blue}{-1}\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto 0 - \color{blue}{\left(\frac{y}{z} \cdot x + -1 \cdot x\right)} \]

      9. *-commutative N/A

         \[\leadsto 0 - \left(\color{blue}{x \cdot \frac{y}{z}} + -1 \cdot x\right) \]

      10. associate-/l* N/A

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot y}{z}} + -1 \cdot x\right) \]

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. --lowering--.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/l* N/A

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

      22. *-lowering-*.f64 N/A

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

      23. /-lowering-/.f64 59.3

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

   5. Simplified 59.3%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -2 \cdot 10^{-114}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 10^{-310}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 38.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot \left(y - z\right)}{t - z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-313}:\\ \;\;\;\;t \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;t\_1 \leq 10^{+197}:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (* x_m (- y z)) (- t z))))
   (*
    x_s
    (if (<= t_1 5e-313)
      (* t (/ x_m z))
      (if (<= t_1 1e+197) x_m (* z (/ x_m 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 t_1 = (x_m * (y - z)) / (t - z);
	double tmp;
	if (t_1 <= 5e-313) {
		tmp = t * (x_m / z);
	} else if (t_1 <= 1e+197) {
		tmp = x_m;
	} else {
		tmp = z * (x_m / 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) :: t_1
    real(8) :: tmp
    t_1 = (x_m * (y - z)) / (t - z)
    if (t_1 <= 5d-313) then
        tmp = t * (x_m / z)
    else if (t_1 <= 1d+197) then
        tmp = x_m
    else
        tmp = z * (x_m / 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 t_1 = (x_m * (y - z)) / (t - z);
	double tmp;
	if (t_1 <= 5e-313) {
		tmp = t * (x_m / z);
	} else if (t_1 <= 1e+197) {
		tmp = x_m;
	} else {
		tmp = z * (x_m / 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):
	t_1 = (x_m * (y - z)) / (t - z)
	tmp = 0
	if t_1 <= 5e-313:
		tmp = t * (x_m / z)
	elif t_1 <= 1e+197:
		tmp = x_m
	else:
		tmp = z * (x_m / 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)
	t_1 = Float64(Float64(x_m * Float64(y - z)) / Float64(t - z))
	tmp = 0.0
	if (t_1 <= 5e-313)
		tmp = Float64(t * Float64(x_m / z));
	elseif (t_1 <= 1e+197)
		tmp = x_m;
	else
		tmp = Float64(z * Float64(x_m / 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)
	t_1 = (x_m * (y - z)) / (t - z);
	tmp = 0.0;
	if (t_1 <= 5e-313)
		tmp = t * (x_m / z);
	elseif (t_1 <= 1e+197)
		tmp = x_m;
	else
		tmp = z * (x_m / 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_] := Block[{t$95$1 = N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, 5e-313], N[(t * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+197], x$95$m, N[(z * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 5.00000000002e-313

   1. Initial program 82.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r+ N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. div-sub N/A

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

      7. --lowering--.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-out-- N/A

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

      11. *-lowering-*.f64 N/A

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

      12. --lowering--.f64 48.5

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

   5. Simplified 48.5%

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

   6. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 7.5

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

   8. Simplified 7.5%

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

   if 5.00000000002e-313 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 9.9999999999999995e196

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 41.8%

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

   if 9.9999999999999995e196 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 36.3%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{t - z}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. mul-1-neg N/A

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

      10. /-lowering-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. --lowering--.f64 58.1

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

   7. Simplified 58.1%

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

   8. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 55.8

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

   10. Simplified 55.8%

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

Alternative 7: 36.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \left(y - z\right)}{t - z} \leq 5 \cdot 10^{-313}:\\ \;\;\;\;t \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (if (<= (/ (* x_m (- y z)) (- t z)) 5e-313) (* t (/ x_m z)) 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 (((x_m * (y - z)) / (t - z)) <= 5e-313) {
		tmp = t * (x_m / z);
	} 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 (((x_m * (y - z)) / (t - z)) <= 5d-313) then
        tmp = t * (x_m / z)
    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 (((x_m * (y - z)) / (t - z)) <= 5e-313) {
		tmp = t * (x_m / z);
	} 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 ((x_m * (y - z)) / (t - z)) <= 5e-313:
		tmp = t * (x_m / z)
	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 (Float64(Float64(x_m * Float64(y - z)) / Float64(t - z)) <= 5e-313)
		tmp = Float64(t * Float64(x_m / z));
	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 (((x_m * (y - z)) / (t - z)) <= 5e-313)
		tmp = t * (x_m / z);
	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[N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], 5e-313], N[(t * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 5.00000000002e-313

   1. Initial program 82.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r+ N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. div-sub N/A

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

      7. --lowering--.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-out-- N/A

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

      11. *-lowering-*.f64 N/A

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

      12. --lowering--.f64 48.5

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

   5. Simplified 48.5%

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

   6. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 7.5

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

   8. Simplified 7.5%

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

   if 5.00000000002e-313 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 80.2%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 41.6%

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

Alternative 8: 70.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+163}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{+74}:\\ \;\;\;\;z \cdot \frac{x\_m}{z - t}\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{+73}:\\ \;\;\;\;x\_m \cdot \frac{y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -1.25e+163)
    x_m
    (if (<= z -1.8e+74)
      (* z (/ x_m (- z t)))
      (if (<= z 8.6e+73) (* x_m (/ y (- t z))) 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 <= -1.25e+163) {
		tmp = x_m;
	} else if (z <= -1.8e+74) {
		tmp = z * (x_m / (z - t));
	} else if (z <= 8.6e+73) {
		tmp = x_m * (y / (t - z));
	} 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 <= (-1.25d+163)) then
        tmp = x_m
    else if (z <= (-1.8d+74)) then
        tmp = z * (x_m / (z - t))
    else if (z <= 8.6d+73) then
        tmp = x_m * (y / (t - z))
    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 <= -1.25e+163) {
		tmp = x_m;
	} else if (z <= -1.8e+74) {
		tmp = z * (x_m / (z - t));
	} else if (z <= 8.6e+73) {
		tmp = x_m * (y / (t - z));
	} 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 <= -1.25e+163:
		tmp = x_m
	elif z <= -1.8e+74:
		tmp = z * (x_m / (z - t))
	elif z <= 8.6e+73:
		tmp = x_m * (y / (t - z))
	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 <= -1.25e+163)
		tmp = x_m;
	elseif (z <= -1.8e+74)
		tmp = Float64(z * Float64(x_m / Float64(z - t)));
	elseif (z <= 8.6e+73)
		tmp = Float64(x_m * Float64(y / Float64(t - z)));
	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 <= -1.25e+163)
		tmp = x_m;
	elseif (z <= -1.8e+74)
		tmp = z * (x_m / (z - t));
	elseif (z <= 8.6e+73)
		tmp = x_m * (y / (t - z));
	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, -1.25e+163], x$95$m, If[LessEqual[z, -1.8e+74], N[(z * N[(x$95$m / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.6e+73], N[(x$95$m * N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x$95$m]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.25e163 or 8.60000000000000026e73 < z

   1. Initial program 61.4%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 73.8%

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

   if -1.25e163 < z < -1.79999999999999994e74

   1. Initial program 79.6%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{t - z}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. mul-1-neg N/A

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

      10. /-lowering-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. --lowering--.f64 91.6

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

   7. Simplified 91.6%

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

   if -1.79999999999999994e74 < z < 8.60000000000000026e73

   1. Initial program 94.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 73.5

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

   5. Simplified 73.5%

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

Alternative 9: 60.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+71}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq -9200000000000:\\ \;\;\;\;x\_m \cdot \left(-\frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 1.2:\\ \;\;\;\;y \cdot \frac{x\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -4.2e+71)
    x_m
    (if (<= z -9200000000000.0)
      (* x_m (- (/ y z)))
      (if (<= z 1.2) (* 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 <= -4.2e+71) {
		tmp = x_m;
	} else if (z <= -9200000000000.0) {
		tmp = x_m * -(y / z);
	} else if (z <= 1.2) {
		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 <= (-4.2d+71)) then
        tmp = x_m
    else if (z <= (-9200000000000.0d0)) then
        tmp = x_m * -(y / z)
    else if (z <= 1.2d0) 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 <= -4.2e+71) {
		tmp = x_m;
	} else if (z <= -9200000000000.0) {
		tmp = x_m * -(y / z);
	} else if (z <= 1.2) {
		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 <= -4.2e+71:
		tmp = x_m
	elif z <= -9200000000000.0:
		tmp = x_m * -(y / z)
	elif z <= 1.2:
		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 <= -4.2e+71)
		tmp = x_m;
	elseif (z <= -9200000000000.0)
		tmp = Float64(x_m * Float64(-Float64(y / z)));
	elseif (z <= 1.2)
		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 <= -4.2e+71)
		tmp = x_m;
	elseif (z <= -9200000000000.0)
		tmp = x_m * -(y / z);
	elseif (z <= 1.2)
		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, -4.2e+71], x$95$m, If[LessEqual[z, -9200000000000.0], N[(x$95$m * (-N[(y / z), $MachinePrecision])), $MachinePrecision], If[LessEqual[z, 1.2], N[(y * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], x$95$m]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -4.19999999999999978e71 or 1.19999999999999996 < z

   1. Initial program 68.8%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 66.3%

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

   if -4.19999999999999978e71 < z < -9.2e12

   1. Initial program 88.9%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. associate-/l* N/A

         \[\leadsto 0 - \color{blue}{x \cdot \frac{y - z}{z}} \]

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \color{blue}{-1}\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto 0 - \color{blue}{\left(\frac{y}{z} \cdot x + -1 \cdot x\right)} \]

      9. *-commutative N/A

         \[\leadsto 0 - \left(\color{blue}{x \cdot \frac{y}{z}} + -1 \cdot x\right) \]

      10. associate-/l* N/A

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot y}{z}} + -1 \cdot x\right) \]

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. --lowering--.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/l* N/A

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

      22. *-lowering-*.f64 N/A

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

      23. /-lowering-/.f64 60.5

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

   5. Simplified 60.5%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{z}} \]
   7. Step-by-step derivation

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. neg-lowering-neg.f64 51.6

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

   8. Simplified 51.6%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. /-rgt-identity N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. neg-lowering-neg.f64 51.6

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

   10. Applied egg-rr 51.6%

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

   if -9.2e12 < z < 1.19999999999999996

   1. Initial program 95.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 97.4

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

   4. Applied egg-rr 97.4%

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

   5. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 75.2

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

   7. Simplified 75.2%

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

   8. Taylor expanded in y around inf

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

   10. Simplified 67.0%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+71}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -9200000000000:\\ \;\;\;\;x \cdot \left(-\frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 1.2:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+71}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq -760000000000:\\ \;\;\;\;-y \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;z \leq 0.014:\\ \;\;\;\;y \cdot \frac{x\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -4e+71)
    x_m
    (if (<= z -760000000000.0)
      (- (* y (/ x_m z)))
      (if (<= z 0.014) (* 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 <= -4e+71) {
		tmp = x_m;
	} else if (z <= -760000000000.0) {
		tmp = -(y * (x_m / z));
	} else if (z <= 0.014) {
		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 <= (-4d+71)) then
        tmp = x_m
    else if (z <= (-760000000000.0d0)) then
        tmp = -(y * (x_m / z))
    else if (z <= 0.014d0) 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 <= -4e+71) {
		tmp = x_m;
	} else if (z <= -760000000000.0) {
		tmp = -(y * (x_m / z));
	} else if (z <= 0.014) {
		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 <= -4e+71:
		tmp = x_m
	elif z <= -760000000000.0:
		tmp = -(y * (x_m / z))
	elif z <= 0.014:
		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 <= -4e+71)
		tmp = x_m;
	elseif (z <= -760000000000.0)
		tmp = Float64(-Float64(y * Float64(x_m / z)));
	elseif (z <= 0.014)
		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 <= -4e+71)
		tmp = x_m;
	elseif (z <= -760000000000.0)
		tmp = -(y * (x_m / z));
	elseif (z <= 0.014)
		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, -4e+71], x$95$m, If[LessEqual[z, -760000000000.0], (-N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]), If[LessEqual[z, 0.014], N[(y * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], x$95$m]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -4.0000000000000002e71 or 0.0140000000000000003 < z

   1. Initial program 68.8%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 66.3%

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

   if -4.0000000000000002e71 < z < -7.6e11

   1. Initial program 88.9%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. associate-/l* N/A

         \[\leadsto 0 - \color{blue}{x \cdot \frac{y - z}{z}} \]

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto 0 - x \cdot \left(\frac{y}{z} + \color{blue}{-1}\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto 0 - \color{blue}{\left(\frac{y}{z} \cdot x + -1 \cdot x\right)} \]

      9. *-commutative N/A

         \[\leadsto 0 - \left(\color{blue}{x \cdot \frac{y}{z}} + -1 \cdot x\right) \]

      10. associate-/l* N/A

          \[\leadsto 0 - \left(\color{blue}{\frac{x \cdot y}{z}} + -1 \cdot x\right) \]

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. --lowering--.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/l* N/A

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

      22. *-lowering-*.f64 N/A

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

      23. /-lowering-/.f64 60.5

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

   5. Simplified 60.5%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{z}} \]
   7. Step-by-step derivation

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. neg-lowering-neg.f64 51.6

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

   8. Simplified 51.6%

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

   if -7.6e11 < z < 0.0140000000000000003

   1. Initial program 95.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 97.4

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

   4. Applied egg-rr 97.4%

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

   5. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 75.2

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

   7. Simplified 75.2%

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

   8. Taylor expanded in y around inf

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

   10. Simplified 67.0%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+71}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -760000000000:\\ \;\;\;\;-y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 0.014:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+73}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+72}:\\ \;\;\;\;x\_m \cdot \frac{y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -4.4e+73) x_m (if (<= z 4.4e+72) (* x_m (/ y (- t z))) 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 <= -4.4e+73) {
		tmp = x_m;
	} else if (z <= 4.4e+72) {
		tmp = x_m * (y / (t - z));
	} 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 <= (-4.4d+73)) then
        tmp = x_m
    else if (z <= 4.4d+72) then
        tmp = x_m * (y / (t - z))
    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 <= -4.4e+73) {
		tmp = x_m;
	} else if (z <= 4.4e+72) {
		tmp = x_m * (y / (t - z));
	} 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 <= -4.4e+73:
		tmp = x_m
	elif z <= 4.4e+72:
		tmp = x_m * (y / (t - z))
	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 <= -4.4e+73)
		tmp = x_m;
	elseif (z <= 4.4e+72)
		tmp = Float64(x_m * Float64(y / Float64(t - z)));
	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 <= -4.4e+73)
		tmp = x_m;
	elseif (z <= 4.4e+72)
		tmp = x_m * (y / (t - z));
	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, -4.4e+73], x$95$m, If[LessEqual[z, 4.4e+72], N[(x$95$m * N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -4.4e73 or 4.4e72 < z

   1. Initial program 65.2%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 72.7%

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

   if -4.4e73 < z < 4.4e72

   1. Initial program 94.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 73.5

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

   5. Simplified 73.5%

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

Alternative 12: 60.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+66}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 250:\\ \;\;\;\;y \cdot \frac{x\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (if (<= z -3.4e+66) x_m (if (<= z 250.0) (* 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 <= -3.4e+66) {
		tmp = x_m;
	} else if (z <= 250.0) {
		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 <= (-3.4d+66)) then
        tmp = x_m
    else if (z <= 250.0d0) 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 <= -3.4e+66) {
		tmp = x_m;
	} else if (z <= 250.0) {
		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 <= -3.4e+66:
		tmp = x_m
	elif z <= 250.0:
		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 <= -3.4e+66)
		tmp = x_m;
	elseif (z <= 250.0)
		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 <= -3.4e+66)
		tmp = x_m;
	elseif (z <= 250.0)
		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, -3.4e+66], x$95$m, If[LessEqual[z, 250.0], N[(y * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.4000000000000003e66 or 250 < z

   1. Initial program 69.7%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 65.9%

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

   if -3.4000000000000003e66 < z < 250

   1. Initial program 94.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 96.5

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

   4. Applied egg-rr 96.5%

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

   5. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 71.4

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

   7. Simplified 71.4%

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

   8. Taylor expanded in y around inf

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

   10. Simplified 61.4%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+66}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 250:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.0% accurate, 0.8× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (/ x_m (/ (- t z) (- y 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 / ((t - z) / (y - 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 / ((t - z) / (y - 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 / ((t - z) / (y - 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 / ((t - z) / (y - 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(t - z) / Float64(y - 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 / ((t - z) / (y - 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[(t - z), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.6%

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

   1. associate-/l* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. --lowering--.f64 N/A

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

   7. --lowering--.f64 98.2

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

4. Applied egg-rr 98.2%

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

Alternative 14: 35.5% accurate, 23.0× 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 #s(literal 1 binary64) 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 81.6%

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

3. Taylor expanded in z around inf

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

5. Simplified 40.6%

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

Developer Target 1: 97.0% accurate, 0.8× 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 2024204 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (/ x (/ (- t z) (- y z))))

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