Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.7 → 0.2

Time: 9.8s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \frac{x \cdot t_0}{z}\\ \mathbf{if}\;z \leq -1.609460262949987 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.0468777931324784 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (<= z -1.609460262949987e-22)
     t_1
     (if (<= z 1.0468777931324784e-7) (/ x (/ z t_0)) 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 (z <= -1.609460262949987e-22) {
		tmp = t_1;
	} else if (z <= 1.0468777931324784e-7) {
		tmp = x / (z / t_0);
	} 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 (z <= (-1.609460262949987d-22)) then
        tmp = t_1
    else if (z <= 1.0468777931324784d-7) then
        tmp = x / (z / t_0)
    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 (z <= -1.609460262949987e-22) {
		tmp = t_1;
	} else if (z <= 1.0468777931324784e-7) {
		tmp = x / (z / t_0);
	} 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 z <= -1.609460262949987e-22:
		tmp = t_1
	elif z <= 1.0468777931324784e-7:
		tmp = x / (z / t_0)
	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 (z <= -1.609460262949987e-22)
		tmp = t_1;
	elseif (z <= 1.0468777931324784e-7)
		tmp = Float64(x / Float64(z / t_0));
	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 (z <= -1.609460262949987e-22)
		tmp = t_1;
	elseif (z <= 1.0468777931324784e-7)
		tmp = x / (z / t_0);
	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[z, -1.609460262949987e-22], t$95$1, If[LessEqual[z, 1.0468777931324784e-7], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

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 2 regimes

2. if z < -1.6094602629499871e-22 or 1.04687779313247842e-7 < z

   1. Initial program 0.1

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

   if -1.6094602629499871e-22 < z < 1.04687779313247842e-7

   1. Initial program 6.2

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

   2. Simplified 16.2

      \[\leadsto \color{blue}{\frac{\sin y}{z \cdot \frac{y}{x}}} \]
      Proof
      (/.f64 (sin.f64 y) (*.f64 z (/.f64 y x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (sin.f64 y) (/.f64 y x)) z)): 18 points increase in error, 48 points decrease in error
      (/.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (sin.f64 y) y) x)) z): 9 points increase in error, 50 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x (/.f64 (sin.f64 y) y))) z): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 13.3

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

   4. Applied egg-rr 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.609460262949987 \cdot 10^{-22}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{elif}\;z \leq 1.0468777931324784 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.9
Cost	13508

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t_0 \leq 10^{-298}:\\ \;\;\;\;\frac{\sin y}{\frac{z}{\frac{x}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \]

Alternative 2
Error	2.8
Cost	13508

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t_0 \leq 10^{-298}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \]

Alternative 3
Error	3.5
Cost	7112

\[\begin{array}{l} t_0 := \frac{\sin y}{\frac{y}{\frac{x}{z}}}\\ \mathbf{if}\;y \leq -0.007115263252903328:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.6734385262702222 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	2.9
Cost	6848

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

Alternative 5
Error	23.1
Cost	712

\[\begin{array}{l} t_0 := \frac{y}{\frac{z}{\frac{x}{y}}}\\ \mathbf{if}\;y \leq -4418103140356633.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.842630226324185 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	23.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -4418103140356633.5:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq 6.842630226324185 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{y}}}\\ \end{array} \]

Alternative 7
Error	23.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -4418103140356633.5:\\ \;\;\;\;\frac{y}{\frac{z \cdot y}{x}}\\ \mathbf{elif}\;y \leq 6.842630226324185 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{y}}}\\ \end{array} \]

Alternative 8
Error	23.1
Cost	712

\[\begin{array}{l} t_0 := \left(\frac{x}{z} + 1\right) + -1\\ \mathbf{if}\;y \leq -85.54204582607974:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.2862554996485113 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	28.5
Cost	320

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

Alternative 10
Error	28.3
Cost	192

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

Reproduce

herbie shell --seed 2022297 
(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))