Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.5 → 0.8

Time: 11.0s

Precision: binary64

Cost: 7113

Math

\[\frac{x \cdot \frac{\sin y}{y}}{z} \]

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-134} \lor \neg \left(z \leq 4.2 \cdot 10^{-120}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.5e-134) (not (<= z 4.2e-120)))
   (* (/ (sin y) y) (/ x z))
   (/ x (* y (/ z (sin y))))))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.5e-134) || !(z <= 4.2e-120)) {
		tmp = (sin(y) / y) * (x / z);
	} else {
		tmp = x / (y * (z / sin(y)));
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-2.5d-134)) .or. (.not. (z <= 4.2d-120))) then
        tmp = (sin(y) / y) * (x / z)
    else
        tmp = x / (y * (z / sin(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return (x * (Math.sin(y) / y)) / z;
}

↓

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.5e-134) || !(z <= 4.2e-120)) {
		tmp = (Math.sin(y) / y) * (x / z);
	} else {
		tmp = x / (y * (z / Math.sin(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	return (x * (math.sin(y) / y)) / z

↓

def code(x, y, z):
	tmp = 0
	if (z <= -2.5e-134) or not (z <= 4.2e-120):
		tmp = (math.sin(y) / y) * (x / z)
	else:
		tmp = x / (y * (z / math.sin(y)))
	return tmp

Julia

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

↓

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.5e-134) || !(z <= 4.2e-120))
		tmp = Float64(Float64(sin(y) / y) * Float64(x / z));
	else
		tmp = Float64(x / Float64(y * Float64(z / sin(y))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.5e-134) || ~((z <= 4.2e-120)))
		tmp = (sin(y) / y) * (x / z);
	else
		tmp = x / (y * (z / sin(y)));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := If[Or[LessEqual[z, -2.5e-134], N[Not[LessEqual[z, 4.2e-120]], $MachinePrecision]], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(z / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	2.5
Target	0.2
Herbie	0.8

\[\begin{array}{l} \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if z < -2.5000000000000002e-134 or 4.2000000000000001e-120 < z

   1. Initial program 0.9

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]

   2. Simplified 3.9

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

      [Start] 0.9	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      associate-/l* [=>] 3.9	\[ \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

   3. Applied egg-rr 1.0

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

   if -2.5000000000000002e-134 < z < 4.2000000000000001e-120

   1. Initial program 7.4

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]

   2. Simplified 0.3

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

      [Start] 7.4	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      associate-/l* [=>] 0.2	\[ \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
      associate-/r/ [=>] 0.3	\[ \frac{x}{\color{blue}{\frac{z}{\sin y} \cdot y}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-134} \lor \neg \left(z \leq 4.2 \cdot 10^{-120}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.0
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-5} \lor \neg \left(y \leq 5 \cdot 10^{-20}\right):\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 2
Error	3.5
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq 1.56 \cdot 10^{+38} \lor \neg \left(x \leq 3.9 \cdot 10^{+186}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \end{array} \]

Alternative 3
Error	22.3
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -6.2 \lor \neg \left(y \leq 3.1 \cdot 10^{+36}\right):\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 4
Error	22.3
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -2.9 \lor \neg \left(y \leq 2.4\right):\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 5
Error	23.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+120} \lor \neg \left(y \leq 3.5 \cdot 10^{-36}\right):\\ \;\;\;\;y \cdot \frac{x}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 6
Error	22.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+45} \lor \neg \left(y \leq 2.7 \cdot 10^{+54}\right):\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 7
Error	28.3
Cost	320

\[\frac{1}{\frac{z}{x}} \]

Alternative 8
Error	28.1
Cost	192

\[\frac{x}{z} \]

Reproduce

herbie shell --seed 2023023 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< z -4.2173720203427147e-29) (/ (* x (/ 1.0 (/ y (sin y)))) z) (if (< z 4.446702369113811e+64) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1.0 (/ y (sin y)))) z)))

  (/ (* x (/ (sin y) y)) z))