Average Error: 14.2 → 0.1
Time: 8.1s
Precision: binary64
Cost: 6720
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
↓
\[\frac{\tan \left(x \cdot 0.5\right)}{0.75} \]
(FPCore (x)
 :precision binary64
 (/ (* (* (/ 8.0 3.0) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))
↓
(FPCore (x) :precision binary64 (/ (tan (* x 0.5)) 0.75))
double code(double x) {
	return (((8.0 / 3.0) * sin((x * 0.5))) * sin((x * 0.5))) / sin(x);
}
↓
double code(double x) {
	return tan((x * 0.5)) / 0.75;
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((8.0d0 / 3.0d0) * sin((x * 0.5d0))) * sin((x * 0.5d0))) / sin(x)
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    code = tan((x * 0.5d0)) / 0.75d0
end function

public static double code(double x) {
	return (((8.0 / 3.0) * Math.sin((x * 0.5))) * Math.sin((x * 0.5))) / Math.sin(x);
}
↓
public static double code(double x) {
	return Math.tan((x * 0.5)) / 0.75;
}

def code(x):
	return (((8.0 / 3.0) * math.sin((x * 0.5))) * math.sin((x * 0.5))) / math.sin(x)
↓
def code(x):
	return math.tan((x * 0.5)) / 0.75

function code(x)
	return Float64(Float64(Float64(Float64(8.0 / 3.0) * sin(Float64(x * 0.5))) * sin(Float64(x * 0.5))) / sin(x))
end
↓
function code(x)
	return Float64(tan(Float64(x * 0.5)) / 0.75)
end

function tmp = code(x)
	tmp = (((8.0 / 3.0) * sin((x * 0.5))) * sin((x * 0.5))) / sin(x);
end
↓
function tmp = code(x)
	tmp = tan((x * 0.5)) / 0.75;
end

code[x_] := N[(N[(N[(N[(8.0 / 3.0), $MachinePrecision] * N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / 0.75), $MachinePrecision]

\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}
↓
\frac{\tan \left(x \cdot 0.5\right)}{0.75}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.2
Target0.3
Herbie0.1
\[\frac{\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

Derivation

  1. Initial program 14.2

    \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

  2. Simplified14.2

    \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{2.6666666666666665}{\sin x}} \]
    Proof
    (*.f64 (*.f64 (sin.f64 (*.f64 x 1/2)) (sin.f64 (*.f64 x 1/2))) (/.f64 8/3 (sin.f64 x))): 0 points increase in error, 0 points decrease in error
    (*.f64 (*.f64 (sin.f64 (*.f64 x 1/2)) (sin.f64 (*.f64 x 1/2))) (/.f64 (Rewrite<= metadata-eval (/.f64 8 3)) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (/.f64 8 3) (sin.f64 x)) (*.f64 (sin.f64 (*.f64 x 1/2)) (sin.f64 (*.f64 x 1/2))))): 0 points increase in error, 0 points decrease in error
    (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (/.f64 8 3) (*.f64 (sin.f64 (*.f64 x 1/2)) (sin.f64 (*.f64 x 1/2)))) (sin.f64 x))): 30 points increase in error, 41 points decrease in error
    (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 8 3) (sin.f64 (*.f64 x 1/2))) (sin.f64 (*.f64 x 1/2)))) (sin.f64 x)): 27 points increase in error, 33 points decrease in error

  3. Applied egg-rr29.7

    \[\leadsto \color{blue}{\frac{1 - \cos x}{\left(\sin x \cdot 0.375\right) \cdot 2}} \]

  4. Applied egg-rr1.0

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan \left(\frac{x}{2}\right)}{0.75}\right)\right)} \]

  5. Applied egg-rr0.1

    \[\leadsto \color{blue}{\frac{\tan \left(x \cdot 0.5\right)}{0.75}} \]

  6. Final simplification0.1

    \[\leadsto \frac{\tan \left(x \cdot 0.5\right)}{0.75} \]

Alternatives

Alternative 1
Error31.9
Cost6656
\[{\left(\frac{1.5}{x}\right)}^{-1} \]
Alternative 2
Error31.9
Cost320
\[2.6666666666666665 \cdot \left(x \cdot 0.25\right) \]
Alternative 3
Error31.9
Cost192
\[x \cdot 0.6666666666666666 \]

Error

Reproduce

herbie shell --seed 2022295 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

  :herbie-target
  (/ (/ (* 8.0 (sin (* x 0.5))) 3.0) (/ (sin x) (sin (* x 0.5))))

  (/ (* (* (/ 8.0 3.0) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))