Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Accuracy: 95.8% → 99.7%

Time: 14.0s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;z \leq -2 \cdot 10^{-44}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y}{\sin y}}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= z -2e-44)
     (/ (/ x z) (/ y (sin y)))
     (if (<= z 7.8e-13) (/ x (/ z t_0)) (/ (* x t_0) z)))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (z <= -2e-44) {
		tmp = (x / z) / (y / sin(y));
	} else if (z <= 7.8e-13) {
		tmp = x / (z / t_0);
	} else {
		tmp = (x * t_0) / z;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (z <= (-2d-44)) then
        tmp = (x / z) / (y / sin(y))
    else if (z <= 7.8d-13) then
        tmp = x / (z / t_0)
    else
        tmp = (x * t_0) / z
    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 t_0 = Math.sin(y) / y;
	double tmp;
	if (z <= -2e-44) {
		tmp = (x / z) / (y / Math.sin(y));
	} else if (z <= 7.8e-13) {
		tmp = x / (z / t_0);
	} else {
		tmp = (x * t_0) / z;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if z <= -2e-44:
		tmp = (x / z) / (y / math.sin(y))
	elif z <= 7.8e-13:
		tmp = x / (z / t_0)
	else:
		tmp = (x * t_0) / z
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (z <= -2e-44)
		tmp = Float64(Float64(x / z) / Float64(y / sin(y)));
	elseif (z <= 7.8e-13)
		tmp = Float64(x / Float64(z / t_0));
	else
		tmp = Float64(Float64(x * t_0) / z);
	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)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (z <= -2e-44)
		tmp = (x / z) / (y / sin(y));
	elseif (z <= 7.8e-13)
		tmp = x / (z / t_0);
	else
		tmp = (x * t_0) / z;
	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_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[z, -2e-44], N[(N[(x / z), $MachinePrecision] / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.8e-13], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision]]]]

Target

Original	95.8%
Target	99.6%
Herbie	99.7%

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

Derivation

1. Split input into 3 regimes

2. if z < -1.99999999999999991e-44

   1. Initial program 99.8%

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

   2. Simplified 92.5%

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

      [Start] 99.8	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      associate-*r/ [<=] 93.1	\[ \color{blue}{x \cdot \frac{\frac{\sin y}{y}}{z}} \]
      associate-/r* [<=] 92.5	\[ x \cdot \color{blue}{\frac{\sin y}{y \cdot z}} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot 1}{\frac{y}{\sin y}}} \]
      Proof

      [Start] 92.5	\[ x \cdot \frac{\sin y}{y \cdot z} \]
      associate-*r/ [=>] 87.6	\[ \color{blue}{\frac{x \cdot \sin y}{y \cdot z}} \]
      *-commutative [=>] 87.6	\[ \frac{x \cdot \sin y}{\color{blue}{z \cdot y}} \]
      times-frac [=>] 99.7	\[ \color{blue}{\frac{x}{z} \cdot \frac{\sin y}{y}} \]
      clear-num [=>] 99.7	\[ \frac{x}{z} \cdot \color{blue}{\frac{1}{\frac{y}{\sin y}}} \]
      associate-*r/ [=>] 99.7	\[ \color{blue}{\frac{\frac{x}{z} \cdot 1}{\frac{y}{\sin y}}} \]

   if -1.99999999999999991e-44 < z < 7.80000000000000009e-13

   1. Initial program 90.0%

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

   2. Simplified 99.7%

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

      [Start] 90.0	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      associate-/l* [=>] 99.7	\[ \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

   if 7.80000000000000009e-13 < z

   1. Initial program 99.8%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-44}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y}{\sin y}}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	20425

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := x \cdot t_0\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-246} \lor \neg \left(t_1 \leq 2 \cdot 10^{-184}\right):\\ \;\;\;\;\frac{t_1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \]

Alternative 2
Accuracy	95.6%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -2.35 \cdot 10^{-8} \lor \neg \left(y \leq 1.55 \cdot 10^{-9}\right):\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 3
Accuracy	98.1%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+148} \lor \neg \left(z \leq 2.35 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{\sin y}{y \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

Alternative 4
Accuracy	95.7%
Cost	6848

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

Alternative 5
Accuracy	95.6%
Cost	6848

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

Alternative 6
Accuracy	65.2%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+26} \lor \neg \left(y \leq 2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(\left(z \cdot y\right) \cdot 0.16666666666666666 + \frac{z}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)}{z}\\ \end{array} \]

Alternative 7
Accuracy	64.1%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+18} \lor \neg \left(y \leq 2.2 \cdot 10^{-19}\right):\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 + y \cdot \left(y \cdot -0.16666666666666666\right)}{z}\\ \end{array} \]

Alternative 8
Accuracy	64.3%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{+18} \lor \neg \left(y \leq 2.2 \cdot 10^{-19}\right):\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + \left(y \cdot y\right) \cdot -0.16666666666666666}}\\ \end{array} \]

Alternative 9
Accuracy	64.3%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+17} \lor \neg \left(y \leq 2.2 \cdot 10^{-19}\right):\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)}{z}\\ \end{array} \]

Alternative 10
Accuracy	64.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+46} \lor \neg \left(y \leq 2.2 \cdot 10^{-19}\right):\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 11
Accuracy	65.2%
Cost	704

\[\frac{\frac{x}{z}}{1 + \left(y \cdot y\right) \cdot 0.16666666666666666} \]

Alternative 12
Accuracy	56.4%
Cost	320

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

Alternative 13
Accuracy	56.6%
Cost	192

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

Reproduce

herbie shell --seed 2023135 
(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))