Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.7 → 0.3

Time: 9.2s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{x \cdot \sin y}{y}}{z}\\ \mathbf{if}\;x \leq -6.935184062160428 \cdot 10^{+78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.0774811368195138 \cdot 10^{-39}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (/ (* x (sin y)) y) z)))
   (if (<= x -6.935184062160428e+78)
     t_0
     (if (<= x 2.0774811368195138e-39) (* (/ x z) (/ (sin y) y)) 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 = ((x * sin(y)) / y) / z;
	double tmp;
	if (x <= -6.935184062160428e+78) {
		tmp = t_0;
	} else if (x <= 2.0774811368195138e-39) {
		tmp = (x / z) * (sin(y) / y);
	} else {
		tmp = 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) :: tmp
    t_0 = ((x * sin(y)) / y) / z
    if (x <= (-6.935184062160428d+78)) then
        tmp = t_0
    else if (x <= 2.0774811368195138d-39) then
        tmp = (x / z) * (sin(y) / y)
    else
        tmp = 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 = ((x * Math.sin(y)) / y) / z;
	double tmp;
	if (x <= -6.935184062160428e+78) {
		tmp = t_0;
	} else if (x <= 2.0774811368195138e-39) {
		tmp = (x / z) * (Math.sin(y) / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = ((x * math.sin(y)) / y) / z
	tmp = 0
	if x <= -6.935184062160428e+78:
		tmp = t_0
	elif x <= 2.0774811368195138e-39:
		tmp = (x / z) * (math.sin(y) / y)
	else:
		tmp = 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(Float64(Float64(x * sin(y)) / y) / z)
	tmp = 0.0
	if (x <= -6.935184062160428e+78)
		tmp = t_0;
	elseif (x <= 2.0774811368195138e-39)
		tmp = Float64(Float64(x / z) * Float64(sin(y) / y));
	else
		tmp = 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 = ((x * sin(y)) / y) / z;
	tmp = 0.0;
	if (x <= -6.935184062160428e+78)
		tmp = t_0;
	elseif (x <= 2.0774811368195138e-39)
		tmp = (x / z) * (sin(y) / y);
	else
		tmp = 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[(N[(N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[x, -6.935184062160428e+78], t$95$0, If[LessEqual[x, 2.0774811368195138e-39], N[(N[(x / z), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	2.7
Target	0.3
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if x < -6.93518406216042771e78 or 2.07748113681951383e-39 < x

   1. Initial program 0.3

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

   2. Taylor expanded in x around 0 0.4

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

   if -6.93518406216042771e78 < x < 2.07748113681951383e-39

   1. Initial program 4.6

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

   2. Applied egg-rr 0.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.935184062160428 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{x \cdot \sin y}{y}}{z}\\ \mathbf{elif}\;x \leq 2.0774811368195138 \cdot 10^{-39}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \sin y}{y}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.9
Cost	7112

\[\begin{array}{l} t_0 := \frac{\sin y}{y \cdot \frac{z}{x}}\\ \mathbf{if}\;y \leq -2.4027625084952013 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.4982128662202893 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	2.9
Cost	7112

\[\begin{array}{l} t_0 := \frac{\sin y \cdot \frac{x}{z}}{y}\\ \mathbf{if}\;y \leq -2.4027625084952013 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8.682737574543665 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	3.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4027625084952013 \cdot 10^{-6}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{elif}\;y \leq 8.682737574543665 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y \cdot \frac{x}{z}}{y}\\ \end{array} \]

Alternative 4
Error	1.6
Cost	7112

\[\begin{array}{l} t_0 := \frac{\frac{x \cdot \sin y}{y}}{z}\\ \mathbf{if}\;x \leq -1.137103673587321 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.5372186697153213 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sin y}{y \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	1.3
Cost	7112

\[\begin{array}{l} t_0 := \frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{if}\;x \leq -2.5230083274581777 \cdot 10^{-104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.624967720134592 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sin y}{y \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	22.7
Cost	968

\[\begin{array}{l} t_0 := y \cdot \left(\left(1 + \frac{x}{y \cdot z}\right) + -1\right)\\ \mathbf{if}\;y \leq -15323595662.813234:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.81577892664334 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	22.9
Cost	712

\[\begin{array}{l} t_0 := \left(\frac{x}{z} + 1\right) + -1\\ \mathbf{if}\;y \leq -15323595662.813234:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.4982128662202893 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	28.3
Cost	192

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

Reproduce

herbie shell --seed 2022298 
(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))