Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.5 → 0.4

Time: 18.9s

Precision: binary64

Cost: 20680

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \frac{x \cdot t_0}{z}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+117}:\\ \;\;\;\;x \cdot \frac{t_0}{z}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (/ (* x t_0) z)))
   (if (<= t_1 -1e+117)
     (* x (/ t_0 z))
     (if (<= t_1 5e-11) (* t_0 (/ x z)) t_1))))

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 t_1 = (x * t_0) / z;
	double tmp;
	if (t_1 <= -1e+117) {
		tmp = x * (t_0 / z);
	} else if (t_1 <= 5e-11) {
		tmp = t_0 * (x / z);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = sin(y) / y
    t_1 = (x * t_0) / z
    if (t_1 <= (-1d+117)) then
        tmp = x * (t_0 / z)
    else if (t_1 <= 5d-11) then
        tmp = t_0 * (x / z)
    else
        tmp = t_1
    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 t_1 = (x * t_0) / z;
	double tmp;
	if (t_1 <= -1e+117) {
		tmp = x * (t_0 / z);
	} else if (t_1 <= 5e-11) {
		tmp = t_0 * (x / z);
	} else {
		tmp = t_1;
	}
	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
	t_1 = (x * t_0) / z
	tmp = 0
	if t_1 <= -1e+117:
		tmp = x * (t_0 / z)
	elif t_1 <= 5e-11:
		tmp = t_0 * (x / z)
	else:
		tmp = t_1
	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)
	t_1 = Float64(Float64(x * t_0) / z)
	tmp = 0.0
	if (t_1 <= -1e+117)
		tmp = Float64(x * Float64(t_0 / z));
	elseif (t_1 <= 5e-11)
		tmp = Float64(t_0 * Float64(x / z));
	else
		tmp = t_1;
	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;
	t_1 = (x * t_0) / z;
	tmp = 0.0;
	if (t_1 <= -1e+117)
		tmp = x * (t_0 / z);
	elseif (t_1 <= 5e-11)
		tmp = t_0 * (x / z);
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+117], N[(x * N[(t$95$0 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-11], N[(t$95$0 * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Target

Original	2.5
Target	0.3
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -1.00000000000000005e117

   1. Initial program 0.2

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

   2. Simplified 0.3

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

      [Start] 0.2	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      rational.json-simplify-2 [=>] 0.2	\[ \frac{\color{blue}{\frac{\sin y}{y} \cdot x}}{z} \]
      rational.json-simplify-49 [=>] 0.3	\[ \color{blue}{x \cdot \frac{\frac{\sin y}{y}}{z}} \]

   if -1.00000000000000005e117 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 5.00000000000000018e-11

   1. Initial program 3.4

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

   2. Simplified 0.5

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

      [Start] 3.4	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      rational.json-simplify-49 [=>] 0.5	\[ \color{blue}{\frac{\sin y}{y} \cdot \frac{x}{z}} \]

   if 5.00000000000000018e-11 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 0.2

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -1 \cdot 10^{+117}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{y}}{z}\\ \mathbf{elif}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.1
Cost	7112

\[\begin{array}{l} t_0 := x \cdot \frac{\sin y}{y \cdot z}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.3
Cost	7112

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := t_0 \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1.42 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-71}:\\ \;\;\;\;x \cdot \frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	3.0
Cost	6848

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

Alternative 4
Error	23.0
Cost	1096

\[\begin{array}{l} t_0 := \frac{x}{y \cdot \left(z \cdot \left(\frac{1}{y} + y \cdot 0.16666666666666666\right)\right)}\\ \mathbf{if}\;y \leq -2.9 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-109}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	22.7
Cost	1096

\[\begin{array}{l} t_0 := \frac{x}{\left(\frac{1}{y} + 0.16666666666666666 \cdot y\right) \cdot \left(y \cdot z\right)}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-39}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	22.8
Cost	896

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

Alternative 7
Error	22.9
Cost	832

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

Alternative 8
Error	23.2
Cost	776

\[\begin{array}{l} t_0 := -1 + \left(1 - \frac{-x}{z}\right)\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.32:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	26.5
Cost	712

\[\begin{array}{l} t_0 := \frac{z}{z \cdot \frac{z}{x}}\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.32:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	26.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq 1.32:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 11
Error	26.8
Cost	712

\[\begin{array}{l} t_0 := \frac{\frac{y}{z}}{\frac{y}{x}}\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{+181}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	27.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+172}:\\ \;\;\;\;z \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 13
Error	28.3
Cost	192

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

Reproduce

herbie shell --seed 2023074 
(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))