Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.7 → 0.2

Time: 19.5s

Precision: binary64

Cost: 20680

Math

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

↓

\[\begin{array}{l} t_0 := \frac{y}{\sin y}\\ t_1 := \frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 2000:\\ \;\;\;\;\frac{\frac{x}{z}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t_0}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (sin y))) (t_1 (/ (* x (/ (sin y) y)) z)))
   (if (<= t_1 -1e+60)
     t_1
     (if (<= t_1 2000.0) (/ (/ x z) t_0) (/ x (* z t_0))))))

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 = y / sin(y);
	double t_1 = (x * (sin(y) / y)) / z;
	double tmp;
	if (t_1 <= -1e+60) {
		tmp = t_1;
	} else if (t_1 <= 2000.0) {
		tmp = (x / z) / t_0;
	} else {
		tmp = x / (z * t_0);
	}
	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 = y / sin(y)
    t_1 = (x * (sin(y) / y)) / z
    if (t_1 <= (-1d+60)) then
        tmp = t_1
    else if (t_1 <= 2000.0d0) then
        tmp = (x / z) / t_0
    else
        tmp = x / (z * t_0)
    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 = y / Math.sin(y);
	double t_1 = (x * (Math.sin(y) / y)) / z;
	double tmp;
	if (t_1 <= -1e+60) {
		tmp = t_1;
	} else if (t_1 <= 2000.0) {
		tmp = (x / z) / t_0;
	} else {
		tmp = x / (z * t_0);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = y / math.sin(y)
	t_1 = (x * (math.sin(y) / y)) / z
	tmp = 0
	if t_1 <= -1e+60:
		tmp = t_1
	elif t_1 <= 2000.0:
		tmp = (x / z) / t_0
	else:
		tmp = x / (z * t_0)
	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(y / sin(y))
	t_1 = Float64(Float64(x * Float64(sin(y) / y)) / z)
	tmp = 0.0
	if (t_1 <= -1e+60)
		tmp = t_1;
	elseif (t_1 <= 2000.0)
		tmp = Float64(Float64(x / z) / t_0);
	else
		tmp = Float64(x / Float64(z * t_0));
	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 = y / sin(y);
	t_1 = (x * (sin(y) / y)) / z;
	tmp = 0.0;
	if (t_1 <= -1e+60)
		tmp = t_1;
	elseif (t_1 <= 2000.0)
		tmp = (x / z) / t_0;
	else
		tmp = x / (z * t_0);
	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[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+60], t$95$1, If[LessEqual[t$95$1, 2000.0], N[(N[(x / z), $MachinePrecision] / t$95$0), $MachinePrecision], N[(x / N[(z * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	2.7
Target	0.3
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -9.9999999999999995e59

   1. Initial program 0.2

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

   if -9.9999999999999995e59 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 2e3

   1. Initial program 3.8

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

   2. Simplified 6.5

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

      [Start] 3.8	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      rational.json-simplify-2 [=>] 3.8	\[ \frac{\color{blue}{\frac{\sin y}{y} \cdot x}}{z} \]
      rational.json-simplify-49 [=>] 3.8	\[ \color{blue}{x \cdot \frac{\frac{\sin y}{y}}{z}} \]
      rational.json-simplify-47 [=>] 6.5	\[ x \cdot \color{blue}{\frac{\sin y}{y \cdot z}} \]

   3. Applied egg-rr 0.3

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

   if 2e3 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 0.2

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

   2. Simplified 9.9

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

      [Start] 0.2	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      rational.json-simplify-49 [=>] 9.9	\[ \color{blue}{\frac{\sin y}{y} \cdot \frac{x}{z}} \]

   3. Applied egg-rr 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -1 \cdot 10^{+60}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{elif}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq 2000:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	20680

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \frac{x \cdot t_0}{z}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 2000:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \end{array} \]

Alternative 2
Error	2.9
Cost	7112

\[\begin{array}{l} t_0 := x \cdot \frac{\sin y}{y \cdot z}\\ \mathbf{if}\;y \leq -5 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.6
Cost	6980

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;z \leq -4.3 \cdot 10^{-16}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t_0}{z}\\ \end{array} \]

Alternative 4
Error	1.6
Cost	6980

\[\begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \end{array} \]

Alternative 5
Error	2.2
Cost	6980

\[\begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+21}:\\ \;\;\;\;\frac{\sin y}{\frac{y}{\frac{x}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \end{array} \]

Alternative 6
Error	2.8
Cost	6848

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

Alternative 7
Error	22.8
Cost	968

\[\begin{array}{l} t_0 := \left(-y\right) \cdot \frac{1}{\left(-z\right) \cdot \frac{y}{x}}\\ \mathbf{if}\;y \leq -50:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	22.8
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{y \cdot 4}{z \cdot \frac{4}{\frac{x}{y}}}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{1}{\left(-z\right) \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 9
Error	25.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -3.25 \cdot 10^{+77}:\\ \;\;\;\;z \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{y \cdot z}\\ \end{array} \]

Alternative 10
Error	23.0
Cost	712

\[\begin{array}{l} t_0 := \frac{x}{y \cdot z} \cdot y\\ \mathbf{if}\;y \leq -0.0014:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	23.1
Cost	712

\[\begin{array}{l} t_0 := \frac{\frac{x}{y}}{z} \cdot y\\ \mathbf{if}\;y \leq -50:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	22.8
Cost	712

\[\begin{array}{l} t_0 := \frac{y}{\frac{y \cdot z}{x}}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	27.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{y \cdot z}\\ \end{array} \]

Alternative 14
Error	28.6
Cost	320

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

Alternative 15
Error	28.5
Cost	192

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

Reproduce

herbie shell --seed 2023064 
(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))